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# Divergence Theorem—Boundaries and Solids

## Lecture no. 35 from the course: Understanding Multivariable Calculus: Problems, Solutions, and Tips

Taught by Professor Bruce H. Edwards | 29 min | Categories: The Great Courses Plus Online Mathematics Courses

Another hallmark of multivariable calculus, the Divergence theorem, combines flux and triple integrals, just as Green’s theorem combines line and double integrals. Discover the divergence of a fluid, and call upon the gradient vector to define how a surface integral over a boundary can give the volume of a solid.

## 36 Lectures

1

A Visual Introduction to 3-D Calculus

0 of 33 min

2

Functions of Several Variables

0 of 30 min

3

Limits, Continuity, and Partial Derivatives

0 of 30 min

4

Partial Derivatives—One Variable at a Time

0 of 30 min

5

Total Differentials and Chain Rules

0 of 31 min

6

Extrema of Functions of Two Variables

0 of 30 min

7

Applications to Optimization Problems

0 of 30 min

8

Linear Models and Least Squares Regression

0 of 31 min

9

Vectors and the Dot Product in Space

0 of 29 min

10

The Cross Product of Two Vectors in Space

0 of 29 min

11

Lines and Planes in Space

0 of 31 min

12

Curved Surfaces in Space

0 of 31 min

13

Vector-Valued Functions in Space

0 of 31 min

14

Kepler’s Laws—The Calculus of Orbits

0 of 30 min

15

Directional Derivatives and Gradients

0 of 30 min

16

Tangent Planes and Normal Vectors to a Surface

0 of 28 min

17

Lagrange Multipliers—Constrained Optimization

0 of 30 min

18

Applications of Lagrange Multipliers

0 of 29 min

19

Iterated integrals and Area in the Plane

0 of 30 min

20

Double Integrals and Volume

0 of 29 min

21

Double Integrals in Polar Coordinates

0 of 30 min

22

Centers of Mass for Variable Density

0 of 29 min

23

Surface Area of a Solid

0 of 31 min

24

Triple Integrals and Applications

0 of 29 min

25

Triple Integrals in Cylindrical Coordinates

0 of 30 min

26

Triple Integrals in Spherical Coordinates

0 of 29 min

27

Vector Fields—Velocity, Gravity, Electricity

0 of 30 min

28

Curl, Divergence, Line Integrals

0 of 31 min

29

More Line Integrals and Work by a Force Field

0 of 31 min

30

Fundamental Theorem of Line Integrals

0 of 31 min

31

Green’s Theorem—Boundaries and Regions

0 of 30 min

32

Applications of Green’s Theorem

0 of 30 min

33

Parametric Surfaces in Space

0 of 32 min

34

Surface Integrals and Flux Integrals

0 of 31 min

35

Divergence Theorem—Boundaries and Solids

0 of 29 min

36

Stokes’s Theorem and Maxwell's Equations

0 of 33 min

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