# Math 53 videos

### Michael Hutchings

There are a few minor errors or flaws in these videos; the ones I am aware of are listed below. If you notice any more, please let me know so that I can try to correct them in the next update.

### Errata.

• 1.1.1: Starting around 7:30, when discussing the second version of the fundamental theorem of calculus, it would make more sense to label the horizontal axis with "t" instead of "x", and to describe the graph of f as the curve y=f(t). (In math one should always avoid using the same letter to denote two different things!)
• 1.1.2: Around 3:35, when parametrizing a circle, 2pi/3 should be 3pi/2.
• 1.1.5: On the last page, on the lower left, I wrote d^2y/dx^2 > 0, when I meant (and said) d^2y/dx^2 < 0.
• 1.1.6: At the very end, when discussing computing area by integrating around the boundary of the region, the sign is wrong.
• 1.5.1: At the bottom of the second page, there is an arrow over the letter s which should not be there (because s here is a parameter, not a vector).
• 2.4.1: In the example at the very end, there is a cos(x) which should be cos(xy).
• 2.6.1: At the beginning, it should not say "isoceles".
• 2.6.2: Around 8:40, x/2=(lambda/2)y should be x/2=lambda(2y). (The subsequent equations are correct.)
• 3.1.2: At the beginning, it should say R=[a,b]x[x,d] instead of f=[a,b]x[c,d].
• 3.4.5: In the bottom line, second integral, I wrote d theta when I meant (and said) d phi.
• 3.5.4: Around 1:32, in the Jacobian of x,y with respect to r,theta, the partial derivative sign is missing in the denominator.

## Part 1: Geometric preliminaries.

### Section 1.1: Introduction to the course, parametrized curves.

• 1.1.1: Review and introduction. This lecture segment gives a very brief review of some notions from single variable calculus, followed by an introduction to multivariable calculus. Watch video. (16:50)
• 1.1.2: Introduction to parametrized curves. This lecture segment defines a parametrized curve and does an example of how to sketch a parametrized curve by plotting points. Watch video. (13:47)
• 1.1.3: More techniques for sketching parametrized curves. This lecture segment explains how to sketch a parametrized curve by eliminating the parameter and using symmetry. We consider the examples of an ellipse and the astroid. Watch video. (9:11)
• 1.1.4: Slope of a parametrized curve. The lecture segment explains how to calculate the slope of a parametrized curve, and uses this to sketch another example of a parametrized curve. Watch video. (10:59)
• 1.1.5: Example: the cycloid. This lecture segment, starting with a description of the cycloid, writes down a parametrization of the cycloid, then calculates the slope, and finally verifies that the segments of the cycloid between singularities are concave. Watch video. (10:09)
• 1.1.6: Area enclosed by parametrized curves. This lecture segment explains how to compute the area of the region between a parametrized curve with no vertical tangents and the x-axis. As examples we compute the area enclosed by the unit circle and the astroid. Watch video. (12:32)
• 1.1.7: Length of a parametrized curve. This lecture segment explains how to compute the length of a parametrized curve. As examples we compute the length of the unit circle and the astroid. Watch video. (16:19)
• 1.1.8: Area of surface of revolution of a parametrized curve. This lecture segment explains how to compute the area of the surface obtained by rotating a parametrized curve around the x- or y-axis in three-dimensional space. As an example we calculate the area of the unit sphere. Watch video. (7:56)

### Section 1.2: Polar coordinates.

• 1.2.1: Introduction to polar coordinates. This lecture segment introduces polar coordinates and explains how to translate between polar coordinates and Cartesian coordinates. We then explain, with examples, how to graph a curve r=f(theta) by plotting points or by converting to a parametrized curve in Cartesian coordinates. Watch video. (17:46)
• 1.2.2: Slope and area in polar coordinates. This lecture segment explains how to find the slope of r=f(theta) by converting to Cartesian coordinates. Example: calculate slope of r=1-2cos(theta) at theta=pi/2. This lecture segment also explains how to calculate areas in polar coordinates by approximation by pie slices. Watch video. (7:25)
• 1.2.3: An example of computing area in polar coordinates. This lecture segment works out an example of computing area in polar coordinates, namely the area of the inside region of r=1-2cos(theta). Watch video. (5:41)
• 1.2.4: Length in polar coordinates. This lecture segment explains how to calculate the lengths of curves in polar coordinates. Example: r=2sin(theta). Watch video. (7:46)

### Section 1.3: Three-dimensional space, vectors, dot product, cross product.

• 1.3.1. Distance in Euclidean space. This lecture segment states the distance formula in two- and three-dimensional Euclidean space, and discusses how this is essentially an axiom which defines Euclidean geometry. Watch video. (9:09)
• 1.3.2. Introduction to vectors. This lecture segment introduces vectors in two- and three-dimensional space along with some of the most basic vector operations, namely length, addition, and multiplication by scalars. Watch video. (12:42)
• 1.3.3. Dot product. This lecture segment defines the dot product of two vectors, studies its algebraic properties, states its geometric interpretation, and proves the vector version of the Pythagorean theorem. Watch video. (13:25)
• 1.3.4. The geometric interpretation of dot product. This lecture segment gives a detailed explanation of the geometric interpretation of dot product which was stated in the previous lecture segment, using orthogonal projection. Watch video. (5:30)
• 1.3.5. Determinants. This lecture segment reviews determinants in two and three dimensions and their geometric interpretation. Watch video. (9:18)
• 1.3.6. Cross product. This lecture defines the cross product of two vectors, discusses its algebraic properties, and explains its geometric interpretation. Watch video. (12:53)

### Section 1.4: Lines, planes, and quadric surfaces.

• 1.4.1: Lines. This lecture segment explains how to write the equation of a line in three-dimensional space in parametrized form, starting from a point on the line and a tangent vector to the line. This lecture segment also explains how to write the equation of a line or line segment determined by two points. Watch video. (9:43)
• 1.4.2: Planes. This lecture segment explains how to write the equation of a plane, given a point on the plane and a normal vector to the plane. This lecture segment also explains how to find the equation of the plane determined by three non-colinear points. Watch video. (9:07)
• 1.4.3. Quadric surfaces 1: ellipsoid, hyperboloid. This lecture segment introduces quadric surfaces and studies two types of quadric surfaces: ellipsoids and hyperboloids. Watch video. (13:09)
• 1.4.4. Quadric surfaces 2: cone, elliptic paraboloid. This lecture segment introduces two more types of quadric surfaces: cones and elliptic paraboloids. Watch video. (7:09)
• 1.4.5. Quadric surfaces 3: hyperbolic paraboloid. This lecture segment introduces one more type of quadric surface: the hyperbolic paraboloid. Watch video. (6:15)
• 1.4.6. Quadric surfaces 4: quadric cylinders and translations. This lecture segment introduces the last type of quadric surface, namely quadric cylinders. This lecture segment also explains how quadric surfaces with extra linear terms (but no cross terms) are translations of the quadric surfaces we have studied previously. Watch video. (6:48)

### Section 1.5: Vector-valued functions and space curves.

• 1.5.1: Introduction to space curves. This lecture segment defines a parametrized curve in three-dimensional space and explains how to regard it as a vector-valued function. We then define the velocity vector, the tangent line, and the acceleration, and consider some examples. Watch video. (12:31)
• 1.5.2: Calculus of vector-valued functions. This lecture explains how to calculate the length of a space curve, then introduces three versions of the product rule for vector-valued functions and considers an example of curves on the unit sphere, and finally introduces integration of vector-valued functions. Watch video. (10:13)

## Part 2: Differentiation.

### Section 2.1: Functions of several variables; limits and continuity.

• 2.1.1: Functions of two variables. This lecture segment defines a function of two variables and explains how the graph of such a function is a surface in three-dimensional space. Watch video. (6:23)
• 2.1.2: Examples of graphs. This lecture segment sketches some examples of graphs. Watch video. (10:35)
• 2.1.3: Level curves. This lecture segment explains how one can alternately illustrate a function of two variables by drawing its level sets. Watch video. (9:27)
• 2.1.4: Functions of three variables. This lecture segment defines functions of three variables and shows how one can illustrate them by drawing level surfaces. Watch video. (6:25)
• 2.1.5: The definition of limit. This lecture segment presents the definition of a limit of a function of two variables. Watch video. (7:06)
• 2.1.6: Properties of limits and continuous functions. This lecture segment introduces some properties of limits and the definition of continuous functions. These allow one to compute some limits without directly using the epsilon-delta definition. Watch video. (7:43)
• 2.1.7: Proving that limits do not exist. This lecture segment shows how one can sometimes show that the limit of a function of two variables does not exist by considering the limits along different curves. Watch video. (15:15)

### Section 2.2: Partial derivatives, tangent planes, linear approximation.

• 2.2.1: Introduction to partial derivatives. This lecture segment explains the definition of partial derivatives of functions of two variables and how to compute them. Watch video. (9:49)
• 2.2.2: Higher partial derivatives. This lecture segment defines partial derivatives of functions of three variables, defines second and higher partial derivatives, and states Clairaut's theorem. Watch video. (14:03)
• 2.2.3: Implicit partial differentiation. This lecture segment explains how to calculate the partial derivatives of a function which is defined by an implicit equation. We use this method to calculate the partial derivatives of the z coordinate of a point on a sphere. Watch video. (10:34)
• 2.2.4: Tangent planes. This lecture segment explains how to find the tangent plane to the graph of a function of two variables in terms of partial derivatives. As an example we calculate the tangent plane to a sphere. Watch video. (9:06)
• 2.2.5: Linear approximation of functions of one variable. This lecture segment reviews how to use linear approximation to approximate the value of a differentiable function of one variable. Watch video. (5:41)
• 2.2.6: Differentiability. This lecture segment defines what it means for a function of two variables to be "differentiable". The idea is that a function is differentiable at a point if the graph of the function is well approximated by the tangent plane near that point. Watch video. (10:44)
• 2.2.7: Linear approximation of functions of two variables. This lecture segment explains linear approximation of differentiable functions of two variables. Watch video. (6:36)

### Section 2.3: The chain rule.

• 2.3.1: Review of the chain rule in single variable calculus. This lecture segment briefly reviews the chain rule in single variable calculus. Watch video. (3:36)
• 2.3.2: Multivariable chain rule #1. This lecture segment explains the simplest version of the chain rule in multivariable calculus and considers a couple of examples. Watch video. (11:48)
• 2.3.3: More about multivariable chain rule #1. This lecture segment explains the proof of multivariable chain rule #1 and does one more example. Watch video. (6:15)
• 2.3.4: Multivariable chain rule #2. This lecture segment introduces a second multivariable version of the chain rule. Watch video. (5:42)
• 2.3.5: General chain rule. This lecture segment explains the most general version of the chain rule in multivariable calculus. Watch video. (7:11)
• 2.3.6: Implicit partial differentiation revisited. This lecture segment uses the chain rule to clarify how implicit partial differentiation works in general, and states the Implicit Function Theorem. Watch video. (9:12)

### Section 2.4: Directional derivatives and the gradient vector.

• 2.4.1: Definition of directional derivatives and gradient. This lecture segment defines the directional derivatives and gradient of a function of more than one variable, and explains how the directional derivatives of a differentiable function are obtained by taking dot products with the gradient vector. Watch video. (11:36)
• 2.4.2: Properties of the gradient. This lecture segment explains three key properties of the gradient: (1) it points in the direction in which the directional derivative is largest; (2) it can be used to write the first version of the chain rule in vector form; and (3) the gradient is perpendicular to the level sets. Watch video. (9:52)
• 2.4.3: Tangent planes revisited. This lecture segment explains how to find the tangent plane to a level set F(x,y,z)=k, and discusses how this level set is a smooth surface where the gradient is nonzero. Watch video. (8:30)

### Section 2.5: Maxima and minima.

• 2.5.1: Review of optimization in one variable. This lecture segment reviews global and local minima and maxima of functions of one variable, the extreme value theorem, and the second derivative test. Watch video. (12:21)
• 2.5.2: Introduction to optimization in two variables. This lecture segment defines local and global extrema of a function f of two variables defined on a domain D. We prove that for any local extremum (a,b), at least one of the following holds: (1) (a,b) is a critical point of f, (2) f_x and f_y are not both defined at (a,b), (3) (a,b) is on the boundary of D. Watch video. (8:51)
• 2.5.3: Examples of critical points. This lecture segment works out some examples of finding the critical points of functions of two variables, and considers when they are local maxima, local minima, or neither. Watch video. (9:37)
• 2.5.4: The second derivative test. This lecture segment explains the second derivative test for functions of two variables. Watch video. (10:10)
• 2.5.5: An example of the second derivative test. This lecture segment works out an example involving finding and classifying the critical points and extrema of a function of two variables. Watch video. (5:59)
• 2.5.6: The extreme value theorem. This lecture segment states the extreme value theorem for functions of two variables. Watch video. (6:33)
• 2.5.7: Optimization example #1. This lecture segment uses the theory we have developed to find the maximum of a certain function on a square. Watch video. (11:21)
• 2.5.8: Optimization example #2. This lecture segment works out another optimization problem, which is a "word problem" requiring a bit of setup. Watch video. (8:42)

### Section 2.6: Lagrange multipliers.

• 2.6.1: Introduction to optimization with constraints. This lecture segment introduces problems where one has to minimize or maximize a function f(x,y) subject to a constraint g(x,y)=k, and works out a simple example by using the constraint equation to eliminate one of the variables. Watch video. (8:39)
• 2.6.2: Lagrange multipliers. This lecture segment introduces the method of Lagrange multipliers for solving constrained optimization problems, and reworks the previous example using this new method. Watch video. (12:14)
• 2.6.3: More Lagrange multipliers. This lecture segment explains Lagrange multipliers in three dimensions and reworks a previous optimization problem using this method. Watch video. (10:18)
• 2.6.4: A more complicated example of Lagrange multipliers. This lecture segment solves a more complicated example of a Lagrange multiplier problem. Watch video. (8:21)
• 2.6.5: The meaning of Lagrange multipliers. This lecture segment reveals the secret meaning of the Lagrange multiplier lambda. Watch video. (6:12)

## Part 3: Integration.

### Section 3.1: Basics of double integrals.

• 3.1.1: Definition of double integrals over rectangles. This lecture segment first briefly reviews integration in single variable calculus, and then does an analogous construction to define double integrals over rectangles. Watch video. (15:47)
• 3.1.2: How to compute double integrals over rectangles. This lecture segment explains how to compute double integrals over rectangles using Fubini's theorem. Watch video. (17:55)
• 3.1.3: Example of double integration over a rectangle. This lecture segment works out an example of double integration over a rectangle. Watch video. (8:20)
• 3.1.4: Double integrals over more general regions. This lecture segment explains how to define and calculate double integrals over more general regions. Watch video. (12:33)
• 3.1.5: Examples of double integrals over more general regions. This lecture segment works out some examples of integrals over more general regions. Watch video. (16:36)

### Section 3.2: Double integrals in polar coordinates, and surface area

• 3.2.1: Double integrals over polar rectangles. This lecture segment explains how to evaluate double integrals over the analogue of a rectangle in polar coordinates. Watch video. (11:22)
• 3.2.2: Integration in polar coordinates example #1. This lecture segment uses integration in polar coordinates to calculate the volume of the region between the surfaces z=sqrt(x^2+y^2) and z=sqrt(1-x^2-y^2). Watch video. (10:55)
• 3.2.3: Integration in polar coordinates example #2. This lecture segment uses integration in polar coordinates to calculate the area under a bell curve. Watch video. (6:13)
• 3.2.4: Integration in polar coordinates example #3. This lecture segment uses integration in polar coordinates to calculate the volume of a certain kind of ring. Watch video. (9:18)
• 3.2.5: Integration over more general regions in polar coordinates. This lecture segment explains how to evaluate double integrals in polar coordinates over some more general regions, and works out an example. Watch video. (5:03)
• 3.2.6: Surface area. This lecture segment explains how to use double integrals to calculate the area of the graph of a differentiable function of two variables. Watch video. (11:57)

### Section 3.3: Triple integrals.

• 3.3.1: Definition of triple integrals. This lecture segment defines triple integrals and explains how to compute them via iterated integrals and determine the limits of integration. Watch video. (18:08)
• 3.3.2: An example of a triple integral. This lecture segment works out an example of a triple integral, illustrating how one of order of integration can be easier than others. Watch video. (9:37)
• 3.3.3: Changing the order of integration in triple integrals. This lecture segment explains, by means of an example, how one can change the order of integration in a triple integral by logically reasoning through inequalities. Watch video. (8:05)
• 3.3.4: Center of mass. This lecture segment introduces an application of triple integrals, to computing the center of mass of a solid with varying mass density. Watch video. (13:51)

### Section 3.4: Triple integrals in cylindrical and spherical coordinates.

• 3.4.1: Triple integrals in cylindrical coordinates. This lecture segment explains how to evaluate integrals using cylindrical coordinates. Watch video. (6:15)
• 3.4.2: Cylindrical coordinates example #1 This lecture segment works out an example of integration using cylindrical coordinates. Watch video. (7:21)
• 3.4.3: Cylindrical coordinates example #2 This lecture segment works out another example of integration using cylindrical coordinates. Watch video. (7:01)
• 3.4.4: Triple integrals in spherical coordinates This lecture segment explains how to evaluate triple integrals using spherical coordinates. Watch video. (13:15)
• 3.4.5: Spherical coordinates example #1 This lecture segment works out an example of integration using spherical coordinates. Watch video. (4:39)
• 3.4.6: Spherical coordinates example #2 This lecture segment works out another example of integration using spherical coordinates. Watch video. (10:43)

### Section 3.5: Change of variables, Jacobians.

• 3.5.1: Change of variables in single variable calculus. This lecture segment reviews integration by change of variables in single variable calculus, sometimes called "u substitution". Watch video. (9:55)
• 3.5.2: Injections, surjections, and bijections. This lecture segment introduces some terminology regarding functions which will clarify what we are doing when we transform one region into another. Watch video. (6:51)
• 3.5.3: Change of variables for double integrals. This lecture segment states the change of variables formula for double integrals. Watch video. (12:14)
• 3.5.4: Integration in polar coordinates revisited. This lecture segment uses the change of variables formula to rederive the formula for converting a double integral to polar coordinates. Watch video. (3:34)
• 3.5.5: Change of variables for triple integrals. This lecture segment explains change of variables formula for triple integrals, and uses it to rederive the formula for converting a triple integral to spherical coordinates. Watch video. (12:31)
• 3.5.6: Change of variables example #1 This lecture segment works out an example of evaluating a double integral using a change of variables. Watch video. (12:45)
• 3.5.7: Change of variables example #2 This lecture segment works out another example of evaluating a double integral using a change of variables. Watch video. (9:36)

## Part 4: Vector calculus.

### Section 4.1: Vector fields and line integrals.

• 4.1.1: Introduction to vector fields. This lecture segment defines what a vector field is, and introduces the gravitational field as an example. Watch video. (9:03)
• 4.1.2: Conservative vector fields. This lecture segment defines the notion of conservative vector field, and works out some examples of showing that vector fields are or are not conservative. Watch video. (14:50)
• 4.1.3: Line integrals with respect to arc length. This lecture segment introduces three types of line integrals, and explains the first type, integration with respect to arc length, in detail. Watch video. (14:05)
• 4.1.4: Why integration with respect to arc length is well defined. This lecture segment explains why integration with respect to arc length does not depend on the parametrization of the curve, as long as the parametrization does not backtrack. Watch video. (6:53)
• 4.1.5: Line integrals with respect to x or y. This lecture segment defines line integrals with respect to x or y. Watch video. (8:30)
• 4.1.6: Example of integration with respect to x. This lecture segment works out an example of integration with respect to x using different parametrizations. Watch video. (7:51)
• 4.1.7: Line integrals of vector fields. This lecture segment introduces line integrals of vector fields, and also summarizes the three different kinds of line integrals and extends them to three dimensions. Watch video. (11:29)
• 4.1.8: Examples of line integrals of vector fields. This lecture segment works out some examples of line integrals of vector fields. Watch video. (9:32)

### Section 4.2: Fundamental theorem of line integrals.

• 4.2.1: Statement of the fundamental theorem of line integrals. This lecture segment states the fundamental theorem of line integrals, which can be thought of as a generalization of the fundamental theorem of calculus from intervals to parametrized curves. Watch video. (10:43)
• 4.2.2: Proof of the fundamental theorem of line integrals. This lecture segment proves the fundamental theorem of line integrals, using the vector version of the chain rule. Watch video. (5:29)
• 4.2.3: Conservative vector fields and closed curves. This lecture segment defines a closed curve, uses the FTLI to deduce that the integral of a vector field along a closed curve is zero, and discusses the physical interpretation. Watch video. (5:16)
• 4.2.4: A characterization of conservative vector fields. This lecture segment proves that a vector field is conservative if and only if its line integral along every closed curve is zero. Watch video. (14:38)

### Section 4.3: Green's theorem.

• 4.3.1: A more practical characterization of conservative vector fields. This lecture segment states the theorem that a vector field (P,Q) on a simply connected domain in R^2 is conservative if and only if P_y=Q_x. Watch video. (9:45)
• 4.3.2: Statement of Green's theorem. This lecture segment states Green's theorem, after stating the Jordan curve theorem. Watch video. (10:10)
• 4.3.3: Examples of Green's theorem. This lecture segments presents some examples of Green's theorem. Watch video. (13:26)
• 4.3.4: Proof of Green's theorem. This lecture segment explains the proof of Green's theorem. Watch video. (18:02)
• 4.3.5: Proof of the second characterization of conservative vector fields. This lecture segment uses Green's theorem to sketch the proof of the more practical characterization of conservative vector fields on simply connected domains. Watch video. (8:04)
• 4.3.6: An interesting nonconservative vector field. This lecture segment gives an example showing that on a non simply connected domain, a vector field (P,Q) satisfying P_y=Q_x might not be conservative. Watch video. (11:49)

### Section 4.4: Curl and divergence.

• 4.4.1: Review and interpretation of Green's theorem. This lecture segment reviews Green's theorem and discusses its intuitive meaning. Watch video. (7:25)
• 4.4.2: Definition of curl. This lecture segment defines the curl of a vector field in three dimensions and discusses its intuitive meaning. Watch video. (7:38)
• 4.4.3: Curl and conservative vector fields. This lecture segment introduces the fact that a differentiable three-dimensional vector field defined on a simply connected region is conservative if and only if its curl is zero. Watch video. (9:16)
• 4.4.4: Examples of curl and conservative vector fields. This lecture segment works out two examples of determining when three-dimensional vector fields are conservative and finding their potential functions if they exist. Watch video. (9:57)
• 4.4.5: Definition of divergence. This lecture segment defines the divergence of a vector field in three dimensions and discusses its physical interpretation. Watch video. (6:28)
• 4.4.6: Relations between gradient, curl, and divergence. This lecture segment reviews and introduces some new facts about what happens when you apply two of the operations gradient, curl, and divergence in succession. Watch video. (8:52)
• 4.4.7: Alternate statement of Green's theorem using curl. This lecture segment gives a new, equivalent statement of Green's theorem in terms of curl, regarding the domain in the plane as a flat surface in space. Our next goal will be to generalize this to curved surfaces in space. Watch video. (3:59)

### Section 4.5: Parametrized surfaces and surface integrals.

• 4.5.1: Definition of a parametrized surface. This lecture segment introduces the notion of a parametrized surface in three-dimensional space. Watch video. (3:59)
• 4.5.2: Examples of parametrized surfaces. This lecture segment introduces three basic examples of parametrized surfaces. Watch video. (13:18)
• 4.5.3: The tangent plane to a parametrized surface. This lecture segment explains how to find the tangent plane to a parametrized surface at a point on the surface. Watch video. (14:55)
• 4.5.4: Area of parametrized surfaces. This lecture segment explains how to calculate the area of a parametrized surface. Watch video. (11:20)
• 4.5.5: Integration with respect to surface area. This lecture segment introduces integration over a surface with respect to surface area, which is analogous to integration over a curve with respect to arc length. Watch video. (11:45)
• 4.5.6: Orientations of surfaces. This lecture segment defines orientations of surfaces, which we will need in order to integrate vector fields over a surface. Watch video. (13:32)
• 4.5.7: Integration of a vector field over a surface. This lecture segment defines the integral of a vector field over a parametrized surface. Watch video. (10:29)
• 4.5.8: Example of integration of a vector field over a surface. This lecture segment works out an example of integration of a vector field over a surface. Watch video. (8:38)
• 4.5.9: Summary of integrals over curves and surfaces. This lecture segment reviews the three kinds of line integrals and two kinds of surface integrals that we have seen, and how they relate to each other. Watch video. (4:56)

### Section 4.6: Stokes' theorem.

• 4.6.1: Statement of Stokes' theorem. This lecture segment explains the statement of Stokes' theorem and discusses its physical interpretation. Watch video. (5:53)
• 4.6.2: Basic examples of Stokes' theorem. This lecture segment first explains how Green's theorem is a special case of Stokes' theorem, and then uses Stokes' theorem to characterize conservative vector fields in terms of curl in three dimensions. Watch video. (9:50)
• 4.6.3: A computational example using Stokes' theorem. This lecture segment works out an example of using Stokes' theorem to evaluate a difficult line integral of a vector field over a closed curve by instead integrating the curl of the vector field over a surface bounded by the curve. Watch video. (6:49)
• 4.6.4: Proof of Stokes' theorem. This lecture segment explains the proof of Stokes' theorem. Watch video. (16:57)
• 4.6.5: Another computational example of Stokes' theorem. This lecture segment works out another computational example of Stokes's theorem, which is similar to the first but slightly more complicated. Watch video. (10:27)

### Section 4.7: The divergence theorem, conclusion.

• 4.7.1: Statement of the divergence theorem. This lecture states the divergence theorem and discusses its physical interpretation. Watch video. (9:34)
• 4.7.2: An example of the divergence theorem. This lecture segment works out an example in which the divergence theorem is used to relate the fluxes of a vector field across two different surfaces without boundary. Watch video. (12:51)
• 4.7.3: Another example of the divergence theorem. This lecture segment works out an example in which the divergence theorem is used to simply the calculation of the flux across a surface with boundary by using a different surface with the same boundary curve. Watch video. (10:17)
• 4.7.4: Proof of the divergence theorem. This lecture segment explains the proof of the divergence theorem. Watch video. (12:00)
• 4.7.5: Two-dimensional version of the divergence theorem. This lecture segment states a two-dimensional version of the divergence theorem and deduces it from Green's theorem. Watch video. (8:23)
• 4.7.6: Summary of the four main theorems of vector calculus. This lecture segment reviews the four main theorems of vector calculus, all of which relate the integral of a function or vector field over the boundary of a shape to the integral of some kind of derivative over the interior of the shape, and outlines some of the uses of these theorems. Watch video. (10:57)