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Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Resources

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RESOURCE: Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus Resources

Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
Vectors and the Geometry of Space
Vector Functions
Partial Derivatives
Multiple Integrals
Vector Calculus

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Advanced Topics in Single-Variable Calculus and an Introduction to Multivariable Calculus
(continued)
Second-Order Linear Equations

Vectors and the Geometry of Space
•  Three-Dimensional Coordinate Systems
•  Vectors in the Plane
•  Vectors in Three Dimensions
•  The Dot Product
•  The Cross Product
•  Equations of Lines and Planes
•  Cylinders and Quadric Surfaces
•  Cylindrical and Spherical Coordinates
•  Surfaces in Space

Three-Dimensional Coordinate Systems
•  There are many types of coordinate systems, including Cartesian, spherical, and cylindrical coordinates.
•  In the Cartesian system, all three of the parameters are represented as the quantitative distance from the reference plane.

•In order to convert from Cartesian to spherical, you need to convert each parameter separately, as follows: [Equation 1].

Vectors in the Plane
•  In order to adequately describe a plane, you need more than a point–you need its normal vector.
•  The normal vector is perpendicular to the directional vector of the reference point.

•You can find the equation of a vector that describes a plane by using the following equation: [Equation 1] .

Vectors in Three Dimensions
•  Vectors play an important role in physics.
•  In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.
•  The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.
•  Vectors can be added to other vectors according to vector algebra.

The Dot Product
•  The dot product can be found algebraically or geometrically.The algebraic method employs the sum of the products of corresponding parameters, and the geometric method uses the product of the magnitudes of the vectors and the cosine of the angle between them.
•  The dot product is a commutative property.
•  The dot product is a distributive property.

The Cross Product
•  Since the cross product is perpendicular to both original vectors, it is also normal to the plane of the original vectors.
•  If the two original vectors are parallel to each other, the cross product is zero.
•  The cross product can be found both algebraically and geometrically.

Equations of Lines and Planes
•  The slope of the line, and the plane it lies on, is the angle of inclination of that line.
•  A line is a two dimensional representation of a three dimensional geometric object, a plane.
•  The parametric equation of a line can be found using the following equation: < x , y , z > = < xo + ta , yo + tb , zo + tc >x = xo +at;y = yo + bt;z = zo + ct.

Cylinders and Quadric Surfaces
•  A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder.
•  A cylinder can be seen as a polyhedral limiting case of an n-gonal prism where n approaches infinity.
•  A quadric, or quadric surface, is any D-dimensional hypersurface in (D + 1)-dimensional space defined as the locus of zeros of a quadratic polynomial.
•  Cylinder, sphere, ellipsoids, etc.are special cases of quadric surfaces.

Cylindrical and Spherical Coordinates
•A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by three numbers [Equation 1] (See Fig 1 for definitions).

Cylindrical and Spherical Coordinates
•  Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational symmetry about the longitudinal axis.

•A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers [Equation 1] (See Fig 2 for definitions.).

Surfaces in Space
•  To say that a surface is “two-dimensional” means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.
•  The concept of surface finds application in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects.
•  Surfaces could be the locus of zeros of certain functions, usually polynomial functions.

Vector Functions
•  Vector-Valued Functions
•  Arc Length and Speed
•  Calculus of Vector-Valued Functions
•  Arc Length and Curvature
•  Planetary Motion According to Kepler and Newton
•  Tangent Vectors and Normal Vectors

Vector-Valued Functions
•  A vector valued function can be made up of vectors and/or scalars.
•  Each component function in a vector valued function represents the location of the value in a different dimension.
•  The domain of the vector value function is the intersection of the component function domains.
•  Vector valued functions can behave the same ways as vectors, and be evaluated similarly.

Arc Length and Speed
•  Arc length is found by placing a number of points along a curve, connecting them by line segments, and then adding those segment lengths together.
•  The beginning of deriving the formula for arc length starts with Pythagorus’s theorem.
•  When finding the arc length, the integral used needs to be with respect to position, x.
•  When finding the speed along a curve, the integral used needs to be with respect to time, t.

Calculus of Vector-Valued Functions
•  A vector valued function can be made up of vectors and/or scalars.
•  Each component function in a vector valued function represents the location of the value in a different dimension.
•  Vector valued functions can behave the same ways as vectors, and be evaluated similarly.
•  Vector functions are widely used in the study of electromagnetic fields, gravitation fields, and fluid flow.

Arc Length and Curvature
•  The arc length is a function of position, so its derivative will be a function of time.This can give you the rate of change of the position, in relation to time, which is called the curvature.

•The curvature can be found by taking the derivative of the velocity vector, which is given: [Equation 1].

•  This same magnitude can also be found using the concept of calculus, the limit.

Planetary Motion According to Kepler and Newton
•  Kepler’s first law of planetary motion describes the motion of its orbit around the Sun.
•  Kepler’s second law of planetary motion explains the reason why the planet moves faster as it approaches the Sun, and slower as it moves farther away.
•  Kepler’s third law of planetary motion explains how the period of an orbit is related to the semi-major axis of its orbit.
•  Newton takes the information presented by Kepler and uses it to explain that the value of a force on an object is the product of its mass and its orbital acceleration.
•  Newton also clarifies that the orbit is an elliptical shape because while each planet is attracted to the Sun, they are also attracted to each other.

Tangent Vectors and Normal Vectors
•  In order for one vector to be tangent to another vector, the intersection needs to be exactly 90 degrees.On a curve or an uneven object, each point will have a unique normal vector.
•  If you want to check whether two vectors are normal to each other, you can find the dot product of the two and make sure it equals zero.

•If you want to find out exactly what the angle between the two vectors is, you can use the following equation, which also employs the dot product: [Equation 1].

•  In order to find the tangent vector to another vector or object, just take the derivative of the reference vector.

Partial Derivatives
•  Functions of Several Variables
•  Limits and Continuity
•  Partial Derivatives
•  Tangent Planes and Linear Approximations
•  The Chain Rule
•  Directional Derivatives and the Gradient Vector
•  Maximum and Minimum Values
•  Lagrange Multiplers
•  Optimization in Several Variables
•  Applications of Minima and Maxima in Functions of Two Variables

Functions of Several Variables
•  Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom.
•  Unlike a single variable function f(x), for which the limits and continuity of the function need to be checked as x varies on a line (x-axis), multivariable functions have infinite number of paths approaching a single point.
•  In multivariable calculus, gradient, Stokes’, divergence, and Green theorems are specific incarnations of a more general theorem: the generalized Stokes’ theorem.

Limits and Continuity
•The function [Equation 1] has different limit values at the origin, depending on the path taken for the evaluation.

Limits and Continuity
•  Continuity in each argument does not imply multivariate continuity.
•  When taking different paths toward the same point yields different values for the limit, the limit does not exist.

Partial Derivatives
•The partial derivative of a function f with respect to the variable x is variously denoted by [Equation 1].

Partial Derivatives
•  To every point on this surface describing a multi-variable function, there is an infinite number of tangent lines.Partial differentiation is the act of choosing one of these lines and finding its slope.

•As an ordinary derivative, partial derivatives are defined in limit: [Equation 1].

Tangent Planes and Linear Approximations
•For a surface given by a differentiable multivariable function z=f(x,y), the equation of the tangent plane at (x0,y0,z0) is given as [Equation 1].

Tangent Planes and Linear Approximations
•  Since a tangent plane is the best approximation of the surface near the point where the two meet, the tangent plane can be used to approximate the surface near the point.

•The plane describing the linear approximation for a surface described by z=f(x,y) is given as [Equation 1].

The Chain Rule
•  The chain rule can be easily generalized to functions with more than two variables.

•For a single variable functions, the chain rule is a formula for computing the derivative of the composition of two or more functions.For example, the chain rule for f ∘ g (x) ≡ f [g (x)] is [Equation 1].

•  The chain rule can be used when we want to calculate the rate of change of the function U(x,y) as a function of time t, where x=x(t) and y=y(t).

Directional Derivatives and the Gradient Vector
•The directional derivative is defined by the limit [Equation 1] .

Directional Derivatives and the Gradient Vector
•If the function f is differentiable at x, then the directional derivative exists along any vector v, and one gets [Equation 1] .

Directional Derivatives and the Gradient Vector
•  Many of the familiar properties of the ordinary derivative hold for the directional derivative.

Maximum and Minimum Values
•For a function of two variables, the second partial derivative test is based on the sign of [Equation 1] and [Equation 2], where (a,b) is a critical point.

Maximum and Minimum Values
•  There are substantial differences between the functions of one variable and the functions of more than one variable in the identification of global extrema.
•  The maximum and minimum of a function, known collectively as extrema, are the largest and smallest values that the function takes at a point either within a given neighborhood (local or relative extremum) or on the function domain in its entirety (global or absolute extremum).

Lagrange Multiplers
•To maximize f(x,y) subject to g(x,y)=c, we introduce a new variable [Equation 1] , called a Lagrange multiplier, and study the Lagrange function (or Lagrangian) defined by [Equation 2] .

Lagrange Multiplers
•  When the contour line for g = c meets the contour lines of f tangentially do we not increase or decrease the value of f — that is, when the contour lines touch but do not cross.This will often be the situation where a solution to the constrained maximum problem above exists.

•Solve [Equation 1] , and we find a necessary condition for extrema under the given constraint.

Optimization in Several Variables
•  Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives.
•  An optimization process that involves only a single variable is rather straightforward.After finding out the function f(x) to be optimized, local maxima or minima at critical points can easily be found.End points may have maximum/minimum values as well.
•  For a rectangular cuboid shape, given the fixed volume, a cube is the geometric shape that minimizes the surface area.

Applications of Minima and Maxima in Functions of Two Variables
•  The second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
•  To find minima/maxima for functions with two variables, we must first find the first partial derivatives with respect to x and y of the function.

•The function [Equation 1] has a saddle point at (0,-1) and (1,-1), and a local maximum at (3/8, -3.4).

Multiple Integrals
•  Double Integrals Over Rectangles
•  Iterated Integrals
•  Double Integrals Over General Regions
•  Double Integrals in Polar Coordinates
•  Triple Integrals in Cylindrical Coordinates
•  Triple Integrals in Spherical Coordinates
•  Triple Integrals
•  Change of Variables
•  Applications of Multiple Integrals
•  Center of Mass and Inertia

Double Integrals Over Rectangles
•  The multiple integral is a type of definite integral extended to functions of more than one real variable–for example, f(x, y) or f(x, y, z).Integrals of a function of two variables over a region in R2 are called double integrals.
•  The double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three dimensional Cartesian plane where z = f(x, y)) and the plane which contains its domain.
•  If there are more variables than 3, a multiple integral will yield hypervolumes of multi-dimensional functions.

Iterated Integrals
•The function f(x,y), if y is considered a given parameter, can be integrated with respect to x as follows: [Equation 1].

Iterated Integrals
•The result is a function of y and therefore its integral can be considered again.If this is done, the result is the iterated integral [Equation 1].

Iterated Integrals
•It is key to note that this is different, in principle, to the multiple integral [Equation 1].

Double Integrals Over General Regions
•If the domain D is normal with respect to the x-axis, and f : D → R is a continuous function, then α(x) and β(x) (defined on the interval [a, b]) are the two functions that determine D.[Equation 1].

Double Integrals Over General Regions
•  Applying this general method, the projection of <em>D</em> onto either the <em>x</em>-axis or the <em>y</em>-axis should be bounded by the two values, a and b.

•For a domain [Equation 1], we can write the integral over D as[Equation 2].

Double Integrals in Polar Coordinates
•The fundamental relation to make the transformation is the following: [Equation 1].

Double Integrals in Polar Coordinates
•  To switch the integral from Cartesian to polar coordinates, the dx dy differentials in this transformation become ρ dρ dφ.

•Once the function is transformed and the domain evaluated, it is possible to define the formula for the change of variables in polar coordinates: [Equation 1].

Triple Integrals in Cylindrical Coordinates
•Switching from Cartesian to cylindrical coordinates, the transformation of the function is made by the following relation [Equation 1].

Triple Integrals in Cylindrical Coordinates
•  In switching to cylindrical coordinates, the (dx dy dz) differentials in the integral becomes (ρ dρ dφ dz).

•Therefore, an integral evaluated in Cartesian coordinates can be switched to an integral in cylindrical coordinates as[Equation 1].

Triple Integrals in Spherical Coordinates
•Switching from Cartesian to spherical coodinates, the function is transformed by this relation: [Equation 1].

Triple Integrals in Spherical Coordinates
•  For the transformation, the (dx dy dz) differentials in the integral are transformed to (ρ2 sinφ dρ dθ dφ).

•Therefore, an integral evaluated in Cartesian coordinates can be switched to an integral in spherical coordinates as [Equation 1]

Triple Integrals
•  By convention, the triple integral has three integral signs (and a double integral has two integral signs); this is a notational convention which is convenient when computing a multiple integral as an iterated integral.
•  If T is a domain that is normal with respect to the xy-plane and determined by the functions α (x,y) and β(x,y), then .

•To integrate a function with spherical symmetry such as [Equation 1], consider changing integration variable to spherical coordinates.

Change of Variables
•  There exist three main “kinds” of changes of variable (to polar coordinate in R2, and to cylindrical and spherical coordinates in R3); however, more general substitutions can be made using the same principle.
•  When changing integration variables, make sure that the integral domain also changes accordingly.
•  Change of variable should be judiciously applied based on the built-in symmetry of the function to be integrated.

Applications of Multiple Integrals
•Given a set D ⊆ Rn and an integrable function f over D, the average value of f over its domain is given by [Equation 1], where m(D) is the measure of D.

Applications of Multiple Integrals
•The gravitational potential associated with a mass distribution given by a mass measure dm on three-dimensional Euclidean space R3 is [Equation 1].

Applications of Multiple Integrals
•An electric field produced by a distribution of charges given by the volume charge density [Equation 1] is obtained by a triple integral of a vector function: [Equation 2].

Center of Mass and Inertia
•  In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid.

•In the case of a system of particles Pi, i = 1, , n , each with mass mi that are located in space with coordinates ri, i = 1, , n , the coordinates R of the center of mass is given as [Equation 1].

•If the mass distribution is continuous with the density ρ(r) within a volume V, the center of mass is expressed as [Equation 1].

Vector Calculus
•  Vector Fields
•  Conservative Vector Fields
•  Line Integrals
•  Fundamental Theorem for Line Integrals
•  Green’s Theorem
•  Curl and Divergence
•  Parametric Surfaces and Surface Integrals
•  Surface Integrals of Vector Fields
•  Stokes’ Theorem
•  The Divergence Theorem

Vector Fields
•  A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane.
•  Vector fields can be constructed out of scalar fields using the gradient operator.
•  Vector fields can be thought to represent the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (the rate of change of volume of a flow) and curl (the rotation of a flow).

Conservative Vector Fields
•  Conservative vector fields have the following property: The line integral from one point to another is independent of the choice of path connecting the two points; it is path-independent.
•  Conservative vector fields are also irrotational, meaning that (in three dimensions) they have vanishing curl.

•A vector field v is said to be conservative if there exists a scalar field [Equation 1] such that [Equation 2].

Line Integrals
•  The value of the line integral is the sum of the values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).
•  Many simple formulae in physics (for example, W=F·s) have natural continuous analogs in terms of line integrals (W=∫C F· ds).The line integral finds the work done on an object moving through an electric or gravitational field, for example.
•  In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve.

Fundamental Theorem for Line Integrals
•  The gradient theorem implies that line integrals through irrotational vector fields are path-independent.
•  Work done by conservative forces, described by a vector field, does not depend on the path followed by the object, but only the end points, as the above equation shows.
•  Any conservative vector field can be expressed as the gradient of a scalar field.

Green’s Theorem
•  Green’s theorem is a special case of the Kelvin–Stokes theorem, when applied to a region in the xy-plane.
•  Considering only two-dimensional vector fields, Green’s theorem is equivalent to the two-dimensional version of the divergence theorem.
•  Green’s theorem can be used to compute area by line integral.

Curl and Divergence
•  The curl is a vector operator that describes the infinitesimal rotation of a three-dimensional vector field.
•  The direction of the curl is the axis of rotation, as determined by the right-hand rule, and the magnitude of the curl is the magnitude of rotation.
•  Divergence is a vector operator that measures the magnitude of a vector field’s source or sink at a given point in terms of a signed scalar.

Parametric Surfaces and Surface Integrals
•  Parametric representation is the most general way to specify a surface.The curvature and arc length of curves on the surface can both be computed from a given parametrization.
•  The same surface admits many different parametrizations.
•  A surface integral is a definite integral taken over a surface.It can be thought of as the double integral analog of the line integral.

Surface Integrals of Vector Fields
•  The flux is defined as the quantity of fluid flowing through S in unit amount of time.
•  To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, and integrate the obtained field.

•This is expressed as [Equation 1].

Stokes’ Theorem
•  The generalized Stokes’ theorem says that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω.
•  Given a vector field, the Kelvin-Stokes theorem relates the integral of the curl of the vector field over some surface to the line integral of the vector field around the boundary of the surface.The Kelvin–Stokes theorem is a special case of the generalized Stokes’ theorem.
•  By applying the Stokes’ theorem, you can show that the work done by electric field is path-independent.

The Divergence Theorem
•  The divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface.
•  In physics and engineering, the divergence theorem is usually applied in three dimensions.However, it generalizes to any number of dimensions.

•Applying the divergence theorem, we can check that the equation [Equation 1] is nothing but an equation describing Coulomb force written in a differential form.

Second-Order Linear Equations
•  Second-Order Linear Equations
•  Nonhomogeneous Linear Equations
•  Applications of Second-Order Differential Equations
•  Series Solutions

Second-Order Linear Equations
•Linear differential equations are of the form [Equation 1], where [Equation 2].

Second-Order Linear Equations
•  When f(t)=0, the equations are called homogeneous linear differential equations.(Otherwise, the equations are called nonhomogeneous equations).
•  Linear differential equations are differential equations that have solutions which can be added together to form other solutions.

Nonhomogeneous Linear Equations
•  Examples of homogeneous or nonhomogeneous second-order linear differential equation can be found in many different disciplines such as physics, economics, and engineering.
•  In simple cases, for example, where the coefficients A1(t) and A2(t) are constants, the equation can be analytically solved.In general, the solution of the differential equation can only be obtained numerically.
•  Linear differential equations are differential equations that have solutions which can be added together to form other solutions.

Applications of Second-Order Differential Equations
•An ideal spring with a spring constant k is described by the simple harmonic oscillation, whose equation of motion is given in the form of a homogeneous second-order linear differential equation: [Equation 1].

Applications of Second-Order Differential Equations
•  Adding the damping term in the equation of motion, the equation of motion is given as .

•Adding the external force term to the damped harmonic oscillator, we get an nonhomogeneous second-order linear differential equation :[Equation 1].

Series Solutions
•The power series method calls for the construction of a power series solution [Equation 1] for a linear differential equation [Equation 2].

Series Solutions
•  The method assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients.

•Hermit differential equation [Equation 1] has the following power series solution: [Equation 2].

Appendix

Key terms
•  analytic functions a function that is locally given by a convergent power series
•  azimuth an arc of the horizon intercepted between the meridian of the place and a vertical circle passing through the center of any object
•  bijective both injective and surjective
•  bijective both injective and surjective
•  bijective both injective and surjective
•  Cartesian of or pertaining to co-ordinates based on mutually orthogonal axes
•  Cartesian of or pertaining to co-ordinates based on mutually orthogonal axes
•  centroid the point at the center of any shape, sometimes called the center of area or the center of volume
•  chain rule formula for computing the derivative of the composition of two or more functions
•  commutative such that the order in which the operands are taken does not affect their image under the operation
•  conservative force a force with the property that the work done in moving a particle between two points is independent of the path taken
•  continuity lack of interruption or disconnection; the quality of being continuous in space or time

•  contour a line on a map or chart delineating those points which have the same altitude or other plotted quantity: a contour line or isopleth
•  coordinate system a method of representing points in a space of given dimensions by coordinates from an origin
•  critical point a maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined
•  critical point a maximum, minimum, or point of inflection on a curve; a point at which the derivative of a function is zero or undefined
•  cross product also called a vector product; results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them
•  cuboid a parallelepiped having six rectangular faces
•  curl the vector field denoting the rotationality of a given vector field
•  curl the vector field denoting the rotationality of a given vector field
•  curvature the degree to which an objet deviates from being flat
•  cylindrical coordinate  a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis
•  damping the reduction in the magnitude of oscillations by the dissipation of energy
•  derivative a measure of how a function changes as its input changes

•  deterministic having exactly predictable time evolution
•  differentiable having a derivative, said of a function whose domain and co-domain are manifolds
•  differentiable having a derivative, said of a function whose domain and co-domain are manifolds
•  differential an infinitesimal change in a variable, or the result of differentiation
•  differential equation an equation involving the derivatives of a function
•  differential geometry the study of geometry using differential calculus
•  differential geometry the study of geometry using differential calculus
•  divergence  a vector operator that measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar
•  divergence  a vector operator that measures the magnitude of a vector field’s source or sink at a given point, in terms of a signed scalar
•  domain the set of all possible mathematical entities (points) where a given function is defined
•  domain the set of all possible mathematical entities (points) where a given function is defined
•  double integral An integral extended to functions of more than one real variable

•  eccentricity the ratio– constant for any particular conic section– of the distance of a point from the focus to its distance from the directrix
•  electric potential the potential energy per unit charge at a point in a static electric field; voltage
•  electric potential the potential energy per unit charge at a point in a static electric field; voltage
•  ellipse a closed curve; the locus of a point such that the sum of the distances from that point to two other fixed points (called the foci of the ellipse) is constant; equivalently, the conic section that is the intersection of a cone with a plane that does not intersect the base of the cone
•  Euclidean adhering to the principles of traditional geometry, in which parallel lines are equidistant
•  flux the rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux
•  flux the rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux
•  Fubini’s theorem  a result which gives conditions under which it is possible to compute a double integral using iterated integrals
•  Fubini’s theorem  a result which gives conditions under which it is possible to compute a double integral using iterated integrals
•  gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x
•  gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x
•  gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x

•  gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x
•  gradient of a function y = f(x) or the graph of such a function, the rate of change of y with respect to x; that is, the amount by which y changes for a certain (often unit) change in x
•  gravitational constant an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass
•  harmonic oscillator a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement according to Hooke’s law, where k is a positive constant
•  hypersurface a n-dimensional surface in a space (often a Euclidean space) of dimension n+1
•  hypervolume a volume in more than three dimensions
•  intermediate value theorem a statement that claims that, for each value between the least upper bound and greatest lower bound of the image of a continuous function, there is a corresponding point in its domain that the function maps to that value
•  Jacobian determinant  the determinant of the Jacobian matrix
•  Jacobian determinant  the determinant of the Jacobian matrix
•  limit a value to which a sequence or function converges
•  line integral An integral the domain of whose integrand is a curve.
•  linear having the form of a line; straight

•  linearity a relationship between several quantities which can be considered proportional and expressed in terms of linear algebra; any mathematical property of a relationship, operation, or function that is analogous to such proportionality, satisfying additivity and homogeneity
•  manifold a topological space that looks locally like the “ordinary” Euclidean space and is Hausdorff
•  Maxwell’s equations a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits
•  meridian an imaginary great circle on the Earth’s surface, passing through the geographic poles
•  moment of inertia a measure of a body’s resistance to a change in its angular rotation velocity
•  multivariable concerning more than one variable
•  normal a line or vector that is perpendicular to another line, surface, or plane
•  normal a line or vector that is perpendicular to another line, surface, or plane
•  optimization the design and operation of a system or process to make it as good as possible in some defined sense
•  origin the point at which the axes of a coordinate system intersect
•  parallelogram a convex quadrilateral in which each pair of opposite edges is parallel and of equal length
•  parametric of, relating to, or defined using parameters

•  parametrization Is the process of deciding and defining the parameters necessary for a complete or relevant specification of a model or geometric object.
•  perpendicular at or forming a right angle (to)
•  perpendicular at or forming a right angle (to)
•  polar coordinate  a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction
•  polynomial an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power
•  potential energy the energy possessed by an object because of its position (in a gravitational or electric field), or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
•  pseudovector a quantity that transforms like a vector under a proper rotation but gains an additional change of sign under an improper rotation
•  recurrence relation an equation that recursively defines a sequence; each term of the sequence is defined as a function of the preceding terms
•  rigid body an idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects
•  Rolle’s theorem a theorem stating that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative (the slope of the tangent line to the graph of the function) is zero
•  scalar a quantity that has magnitude but not direction; compare vector
•  scalar a quantity that has magnitude but not direction; compare vector

•  scalar function any function whose domain is a vector space and whose value is its scalar field
•  sharpness the fineness of the point a pointed object
•  slope also called gradient; slope or gradient of a line describes its steepness
•  slope also called gradient; slope or gradient of a line describes its steepness
•  spherical coordinate a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith
•  spherical coordinate a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith
•  tangent a straight line touching a curve at a single point without crossing it there
•  tensor a multidimensional array satisfying a certain mathematical transformation
•  vector a directed quantity, one with both magnitude and direction; the signed difference between two points
•  vector a directed quantity, one with both magnitude and direction; the signed difference between two points
•  vector a directed quantity, one with both magnitude and direction; the signed difference between two points
•  vector a directed quantity, one with both magnitude and direction; the signed difference between two points

•  vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
•  vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
•  vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
•  vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
•  vector field a construction in which each point in a Euclidean space is associated with a vector; a function whose range is a vector space
•  velocity a vector quantity that denotes the rate of change of position with respect to time, or a speed with the directional component

Fig 1
Three-Dimensional Space
This is a three dimensional space represented by a Cartesian coordinate system.

A graph of z = x^2 + xy + y^2.
For the partial derivative at (1, 1, 3) that leaves y constant, the corresponding tangent line is parallel to the xz-plane.

Double Integral
Double integral over the normal region D shown in the example.

Volume to be Integrated
Double integral as volume under a surface z = x^2 − y^2.The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.

Cylinder
A right circular cylinder with radius r and height h.

Ellipsoid
An ellipsoid given as x^2 / a^2 + y^2 / b^2 + z^2 / c^2 = 1.

Kelvin-Stokes’ Theorem
An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the “normal” vector n.

Vector-Valued Function
This a graph of a parametric curve (a simple vector-valued function with a single parameter of dimension 1).The graph is of the curve: <2Cos(t), 4Sin(t),t> where t goes from 0 to 8pi.

Spherical Coordinate System
The spherical system is used commonly in mathematics and physics and has variables of r, , and .

Four Most Important Differential Operators
Gradient, curl, divergence, and Laplacian are four most important differential operators.

Damped Harmonic Oscillators
A solution of damped harmonic oscillator.Curves in different colors show various responses depending on the damping ratio.

Maximizing f(x,y)
Find x and y to maximize f(x,y) subject to a constraint (shown in red) g(x,y)=c.

Spherical Coordinate System
Spherical coordinates (r, θ, φ) as often used in mathematics: radial distance r, azimuthal angle θ, and polar angle φ.The meanings of θ and φ have been swapped compared to the physics convention.

Gradient of a Function
The gradient of the function f(x,y) = −[(cosx)^2 + (cosy)^2] depicted as a projected vector field on the bottom plane.Directional derivative represents the rate of change of the function along any direction specified by v.

Spherical Coordinates
Spherical coordinates are useful when domains in R^3 have spherical symmetry.

Scalar Field
The chain rule can be used to take derivatives of multivariable functions with respect to a parameter.

Cylindrical Coordinates
Cylindrical coordinates are often used for integrations on domains with a circular base.

Saddle Point
A saddle point on the graph of z=x2−y2 (in red).

Maclaurin Power Series of an Exponential Function
The exponential function (in blue), and the sum of the first n+1 terms of its Maclaurin power series (in red).Using power series, a linear differential equation of a general form may be solved.

Center of Mass
Two bodies orbiting around the center of mass inside one body

Normal Vector to a Plane
The Plane is described by the vectors and extending from an arbitrary point .

y=1
A slice of the graph in Fig.1 at y= 1

Cylindrical Coordinates
Changing to cylindrical coordinates may be useful depending on the setup of problem.

Curves and the Pythagorean Theorem
For a small piece of curve, ∆s can be approximated with the Pythagorean theorem.

Continuity
Continuity in single variable function as shown is rather obvious.However, continuity in multivariable functions yields many counter-intuitive results.

Graphical Representation of a Triple Integral
Example of domain in R^3 that is normal with respect to the xy-plane.

Vertical Line, Graphed
Vertical line x = a, lying on the x-y plane (z=0).

Example where the contour and constraint cross at an extremum.
Vector valued function
This graph is a visual representation of a three dimensional vector valued function.

Kelvin-Stokes’ Theorem
An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the “normal” vector n.

Simple Pendulum
A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.

Curvature
Curvature is the amount an object deviates from being flat.Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P\.The curvature of C at P is then defined to be the curvature of that circle or line.The radius of curvature is defined as the reciprocal of the curvature.

Fig 1
Kelvin-Stokes’ Theorem
An illustration of the Kelvin–Stokes theorem, with surface Σ, its boundary ∂, and the “normal” vector n.

Heat Transfer
Phenomena, such as heat transfer, can be described using nonhomogeneous second-order linear differential equations.

Computing area by line integral
D is a simple region with its boundary consisting of the curves C1, C2, C3, C4.

Fig 1
The above field v(x,y,z) = (−y/(x2+y2), +x/(x2+y2), 0) includes a vortex at its center, meaning it is non-irrotational; it is neither conservative, nor does it have path independence.However, any simply connected subset that excludes the vortex line (0,0,z) will have zero curl, ∇v = 0.Such vortex-free regions are examples of irrotational vector fields.

Line Integral Over Scalar Field
The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

Figure 1: Normal Vector
These vectors are normal to the plane because the intersection between them and the plane makes a right angle.

A Scalar Field
A scalar field shown as a function of (x,y).Extensions of concepts used for single variable functions may require caution.

Use of an iterated integral
An iterated integral can be used to find the volume of the object in the figure.

The Right Hand Rule
If you use the rules shown in the figure, your thumb will be pointing in the direction of vector c, the cross product of the vectors.

Cylindrical Coordinate System
The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.

The Divergence Theorem
The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, like any of the surfaces on the left.It can not directly be used to calculate the flux through surfaces with boundaries, like those on the right.(Surfaces are blue, boundaries are red.)

A Sphere Defined Parametrically
A sphere can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly (by x² + y² + z² − r² = 0.)

Line Integral Over Scalar Field
The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

Transformation to Polar Coordinates
This figure illustrates graphically a transformation from cartesian to polar coordinates

Cylindrical Coordinate System
A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.

Vector in 3D Space
A vector in the 3D Cartesian space, showing the position of a point A represented by a black arrow.i,j,k are unit vectors in x,y,z axis, respectively

Tangent Plane to a Sphere
The tangent plane to a surface at a given point is the plane that “just touches” the surface at that point.

Arc Length
The arc length is the equivalent of taking a curve, straightening it out, and then measuring it, as seen in this animation.

A Mass to be Integrated
Points x and r, with r contained in the distributed mass (gray) and differential mass dm(r) located at the point r.

Figure 2: Normal Plane
A plane can be determined as normal to the object if the directional vector of the plane makes a right angle with the object at its tangent point.

Plot of z = (x+y)(xy+xy^2)
Plot of z = (x+y)(xy+xy^2).The maxima and minima of this plot cannot be found without extensive calculation.

Rectangular Cuboid
Mathematical Optimization can be used to solve problems that involve finding the right size of a volume such as a cuboid.

Electric Field Lines of a Positive Charge
Electric field lines emanating from a point where positive electric charge is suspended over a negatively charged infinite sheet.Electric field is a vector field which can be represented as a gradient of a scalar field, called electric potential.Therefore, electric force is a conservative force.

Ellipse
The important components of an ellipse are as follows: semi-major axis a, semi-minor axis b, semi-latus rectum p, the center of the ellipse, and its two foci marked by large dots.For θ = 0°, r = rminand for θ = 180°, r = rmax.

The Second Law
Illustration of Kepler’s second law.The planet moves faster near the Sun so that the same area is swept out in a given time as it would be at larger distances, where the planet moves more slowly.The green arrow represents the planet’s velocity, and the purple arrows represent the force on the planet.

Dot Product
When finding the dot product geometrically, you need the magnitudes of the vectors and the angle between them.

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