M344: Calculus III

Section 16.10

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Vector Calculus: A Summary

In this final chapter of the Calculus sequence we've learned about four major theorems that can all be thought of, in some way, as extensions of the Fundamental Theorem of Calculus. In this section we review these theorems, and wrap up our study with a couple of Exercises.

The Fundamental Theorem of Calculus

Let $f$ be a continuous function on an interval $[a,b]$, and let $F$ be any anti-derivative of $f$ on $[a,b]$. Then

1. $\displaystyle \frac{d}{dx}\, \int_a^x f(u)\; du = f(x)$ on $[a,b]$, and
   
   

2. $\displaystyle \int_a^b \dfrac{d}{dx} \Big[ F(x) \Big] \; dx = F(b) - F(a)$.

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The Fundamental Theorem for Path Integrals

Let $C$ be a path oriented from initial point $A$ to terminal point $B$, and let $\mathbf{F}$ be a conservative vector field defined along $C$. If $f$ is any potential function for $\mathbf{F}$, then

$$ \int_C \mathbf{F}\cdot d\mathbf{r} = f(B) - f(A). $$

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Green's Theorem

Let $D$ be a Jordan region and $\mathbf{F} = \langle P, Q \rangle$ be a vector field defined on $D$ such that $P$ and $Q$ have continuous first-order partial derivatives. Then

$$ \int\!\!\!\!\int_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\; dA = \int_C \mathbf{F}\cdot d\mathbf{r}, $$
where $C = \partial D$ is the positively-oriented boundary curve of $D$.

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Stokes' Theorem

Let $S$ be an oriented surface in $\mathbb{R}^3$ with simple closed boundary curve $\partial S = C$ oriented in agreement with the orientation of $S$. Let $\mathbf{F}$ be a vector field defined along $S$ whose component functions have continuous first-order partial derivatives. Then

$$ \int\!\!\!\!\int_S \text{curl}\, \mathbf{F}\cdot d\mathbf{S} = \int_C \mathbf{F}\cdot d\mathbf{r}. $$

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Divergence Theorem

Let $E$ be a simply connected bounded region in $\mathbf{R}^3$, bounded by the positively-oriented surface $\partial E = S$. Suppose $\mathbf{F}$ be a vector field defined on $E$ whose first-order partial derivatives are continuous on $E$. Then

$$ \int\!\!\!\!\int\!\!\!\!\int_E \text{div}\,\mathbf{F}\; dV = \int\!\!\!\!\int_S \mathbf{F}\cdot d\mathbf{S}. $$

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We conclude with a couple of exercises.

1. Exercise

A solid occupies a region $E$ with boundary surface $S$ and is immersed in a liquid with constant density $\rho$. Suppose a coordinate system is set up so that the $xy$-plane coincides with the surface of the liquid, and positive values of $z$ are measured downward into the liquid.

The pressure at depth $d$ is given by $p = \rho gz$ where $g$ is the constant acceleration due to gravity. The total bouyant force on the solid due to the pressure distribution is given by the surface integral,

$$ \mathbf{F} = - \int\!\!\!\!\int_S p\, \mathbf{n}\; dS, $$
where $\mathbf{n}$ is the outward unit normal vector field to $S$.

Show that $\mathbf{F} = - W\,\mathbf{k}$, where $W$ is the weight of the liquid displaced by the solid.

The result is Archimedes Principle: The bouyant force on an object is equal to the weight of the water it displaces.

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2. Exercise

Prove the following identities for triple integrals:

$$ \begin{align}{} \int\!\!\!\!\int\!\!\!\!\int_E \Big( f\,\Delta g + \nabla f \cdot \nabla g \Big) \; dV & = \int\!\!\!\!\int_S \big( f\,\nabla g\big)\cdot d\mathbf{S}, \quad \text{and} \\[3 ex] \int\!\!\!\!\int\!\!\!\!\int_E \Big( f\,\Delta g - g\,\Delta f \Big)\; dV & = \int\!\!\!\!\int_S \Big( f\,\nabla g - g\,\nabla f\Big)\cdot d\mathbf{S}. \end{align} $$

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