Calculus 3 Test #4 Review, Spring 2009

Calculus 3 Test #4 Review SheetPage 1 of 14

Section 14.1: Vector Fields

Definition 1.1: Vector Field

A vector field (in the plane) is a function F(x, y) that maps points in R2 into the set of two-dimensional vectors V2. We write this as:

for scalar functions f1(x, y) and f2(x, y).

In 3D space, a vector field is a function F(x, y, z) that maps points in R3 into the set of three-dimensional vectors V3. We write this as:

for scalar functions f1(x, y, z) and f2(x, y, z), and f3(x, y, z).

We can visualize vector fields by drawing a collection of vectors F(x, y), each with their initial point at (x, y).

Some important vector fields you should know:

(There is a decent vector field graphing tool online at

Uniform vector fields
Example:F(x, y) = 1, 0. /
Radial Vector Fields
Example:F(x, y) = x, y. /
Rotational (Circular) Vector Fields
Example:F(x, y) = -y, x. /

Velocity Fields

Many times vector fields are used to represent the velocity of a particle as a function of its position in space, like the air flow over a car in a wind tunnel test. When this is the case, we can find the path of a particle that starts at any given point (x0, y0) by solving the differential equations:

with initial conditions x(t0) = x0, and y(t0) = y0. The path followed by a particle for a given initial condition is called a flow line.

There is a technique to obtain the equation of a flow line, even if we can not find x(t) and y(t) explicitly. This technique involves using the chain rule to obtain a differential equation for y = f(x).

Let’s say x = g(t) and y = h(t) for some functions g and h that are too hard to compute from the velocity

When the right hand side is separable, we know how to solve this differential equation…

Definition 1.2: Gradient Fields and Potential Functions

The vector field is called gradient field for any scalar function f(x, y). f is also called the potential function for F. Whenever for some scalar function f, we refer to F as a conservative vector field.

Clearly, given any differentiable scalar function , one can always define a corresponding vector field by computing .

However, the converse of this statement is not true. That is, given a vector field F(x, y), it is not always possible to find a corresponding potential function f(x, y) such that (but sometimes it is…).

To find a potential function, either integrate or , making sure to leave your constant of integration as a function of the other variable, then take the partial derivative of your answer with respect to the other variable to find the “constant function” of integration.

Section 14.2: Line Integrals

Definition 2.1: Line Integral with Respect to Arc Length

The line integral of f(x, y, z) with respect to arc length along the oriented curve C in three-dimensional space, written as is defined by

provided the limit exists and is the same for all choices of evaluation points.

Theorem 2.1: Evaluation Theorem (for Line Integrals)

If f(x, y, z) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t), z(t)) for atb, and x(t), y(t), and z(t) have continuous first derivatives, then

Likewise, if f(x, y) is continuous in a region D containing a curve C, C can be described parametrically by (x(t), y(t)) for atb, and x(t) and y(t) have continuous first derivatives, then

A curve C is called smooth if it can be described parametrically by (x(t), y(t), z(t)) for atb, x(t), y(t), and z(t) have continuous first derivatives, and on the interval [a, b].

For example, the curve C = (t2, t3, t2) is not smooth on the interval [-1, 1] because when t = 0:

This means that the line integral is undefined, but if we break up the curve into two subpieces C = C1C2 over the intervals [-1,0] and [0, 1], then we can break up the integral into two subpieces as well. A curve that can be subdivided into smooth subpieces is called piecewise smooth.

Theorem 2.2: Line Integrals on a Piecewise Smooth Curve

If f(x, y, z) is continuous in a region D containing an oriented curve C, and C is piecewise-smooth with C = C1C2 … Cn and all the Ci are smooth and the terminal point of Ci is the initial point of Ci+1 for all i = 1, 2, … n-1, THEN

AND

Geometric interpretation of the line integral:

Just as is the area bounded by x = a, x = b, y = 0 and y = f(x), is the area of the “vertical” surface bounded by the “vertical” line through (x(a), y(a)), the vertical line through (x(b), y(b)), the parametric curve (x(t), y(t), 0), and the parametric curve ( x(t), y(t), f(x(t), y(t)) ).

Theorem 2.3: Arc Length of a Curve from a Line Integral

For any piecewise-smooth curve C, gives the arc length of the curve C.

Next, we worry about evaluating a line integral where the integrand is the dot product of a vector field and the differential tangent vector to the curve, as arises in the computation of work:

Definition 2.2: Component-Wise Line Integrals

The line integral of f(x, y, z) with respect to x along the oriented curve C in three-dimensional space is written as and is defined by:

provided the limit exists and is the same for all choices of evaluation points.

The line integral of f(x, y, z) with respect to y along the oriented curve C in three-dimensional space is written as and is defined by:

provided the limit exists and is the same for all choices of evaluation points.

The line integral of f(x, y, z) with respect to z along the oriented curve C in three-dimensional space is written as and is defined by:

provided the limit exists and is the same for all choices of evaluation points.

Theorem 2.4: Evaluation Theorem (for Component-Wise Line Integrals)

Theorem 2.5: Evaluation Theorem (for Component-Wise Line Integrals)

If f(x, y, z) is continuous in a region D containing an oriented curve C, then:

If C is piecewise-smooth:

If C is piecewise-smooth and forms a closed loop:

Consequence:

Section 14.3: Independence of Path and Conservative Vector Fields

Vocabulary: the line integral is path independent in a domain D if the integral is the same for every path in D that has the same beginning and ending points.

Definition 3.1: Connected Region

A region (for n 2) is called connected if every pair of points in D can be connected by a piece-wise smooth curve lying entirely in D.

Theorem 3.1: Path Independence of the Line Integral of a Conservative Vector Field

If the vector field is continuous on the open, connected region , then the line integral is independent of path in D if and only if F is conservative in D.

Theorem 3.2: Fundamental Theorem for Line Integrals in Two Variables

If the vector field is continuous on the open, connected region , C is any piecewise-smooth curve lying in D with initial point and terminal point , and F is conservative in D with , then

What happens if the curve C starts and ends at the same point? We call a curve Cclosed if its endpoints are the same, which can be described mathematically as:

for .

Theorem 3.3: The Line Integral of a Conservative Vector Field along a Closed Curve is Zero

If the vector field is continuous on the open, connected region , then F is conservative on D if and only if for every piecewise-smooth curve C lying in D.

Here we can see why the word conservative is used to describe these vector fields: if we think of F as a “force field”, then moving an object through the field along a closed path results in no net work being done, so energy is conserved when the object starts and stops at the same point.

Here is a simple test for whether a vector field is conservative:

Theorem 3.4

If and have continuous first partial derivatives on a simply-connected region , then is independent of path in D if and only if for all (x, y) in D.

Conservative Vector Fields
If the vector field is continuous on the open, simply-connected region and and have continuous first partial derivatives on D, then the following five statements are equivalent (i.e., they are either all true or all false)

is conservative in D.
is a gradient field in D. (i.e., ).
is independent of path in D.
foe every piecewise-smooth closed curve C lying in D.
for all (x, y) in D.

Theorem 3.5: Fundamental Theorem for Line Integrals in Three Variables

If the vector field is continuous on the open, connected region , then the line integral is independent of path in D if and only if F is conservative in D. That is, for all (x, y, z) in D for some scalar function f (the potential function for F). Further, any for any piecewise-smooth curve C lying in D with initial point and terminal point , it is true that

Another way to tell if a three-component vector field is conservative is to use partial derivatives like we did with two-component fields:

Remember the last row of equalities when we later learn that another condition for a conservative vector field is if its curl is zero:

Section 14.4: Green's Theorem

Vocabulary:

Curve C:

A curve C is called simple if it does not intersect itself except at the endpoints.

A simple closed curve C has positive orientation if the region R that it bounds “stays to the left” of C as it is traversed.

The notation is used to denote a line integral along a simple closed curve C with a positive orientation.

Theorem 4.1: Green’s Theorem

If C is a piecewise-smooth, simple closed curve in the plane with positive orientation, R is the region enclosed by C together with C, and are continuous with continuous first partial derivatives in some open region D, and RD, then

Vocabulary:

is the contour Cthat bounds a region R. Thus,.

Theorem 4.2:

Suppose and have continuous first partial derivatives on a simply-connected region D. Then is independent of path if and only if for all (x, y) in D.

Section 14.5: Curl and Divergence

Definition 5.1: Curl of a Vector Field

The curl of the vector field is the vector field

defined at all points at which the partial derivatives exist.

Alternative notation:

Definition 5.2: Divergence of a Vector Field

The divergence of the vector field is the scalar function

defined at all points at which the partial derivatives exist.

Alternative notation:

Vocabulary:

irrotational  fluid does not tend to rotate at a given point (curl is zero)

source-free / incompressible  amount that flows in is equal to amount that flows out (divergence is zero)

Theorem 5.1: A Conservative Field is Irrotational

If a conservative vector field has components with continuous first-order partial derivatives throughout an open region , then .

Theorem 5.2: An Irrotational Vector Field is Conservative

If a vector field has components with continuous first-order partial derivatives throughout an open region , then F is conservative if and only if .

Conservative Vector Fields
If the vector field has components with continuous first partial derivatives on , then the following five statements are equivalent (i.e., they are either all true or all false)

is conservative in D.
is a gradient field in D. (i.e., ).
is independent of path in D.
for every piecewise-smooth closed curve C lying in D.
.

Alterative Form of Green’s Theorem:

Section 14.6: Surface Integrals

Definition 6.1: Surface Integral

The surface integral of a function g(x, y, z) over a surface S R3, written as , is defined by

provided the limit exists and is the same for all choices of evaluation points .

Notice that we have to do two things to evaluate this integral:

Write g(x, y, z) as a function of two variables, since a double integral involves only two variables.
Write dS, the differential element of surface area on the surface S, in terms of dA, the differential element of surface area in the x-y plane over which the double integration is performed.

In section 13.4, we found the differential surface element dS for a surface S defined by z = f(x, y) to be .

In this section, the book makes the point that is normal to the plane Ti, and so we can define a normal vector to that small area element . The upshot of this approach is the we can define dS as: , which will be useful for thinking about parametrically defined surfaces. This expression simplifies to.

Theorem 6.1: Evaluation Theorem for Surface Integrals

If a surface S is given by z = f(x, y) for (x, y) in the region R R2, and f has continuous first partial derivatives, then

Parametric Representation of Surfaces

For a surface defined parametrically as , define

and .

ru and rv will be in the tangent plane at any point, which means .

Definition 6.2: Flux

Let be a continuous vector field defined on an oriented surface S with unit normal vector n. The surface integral of F over S (or the flux of S over S) is given by

Section 14.7: The Divergence Theorem

Theorem 7.1: The Divergence Theorem:

If F is a vector field whose components have continuous first partial derivatives over some region bounded by the closed surface Q with an exterior unit normal vector N, then

Section 14.8: Stokes' Theorem

Theorem 8.1: Stokes’s Theorem

If S is an oriented, piecewise-smooth surface with unit normal vector n, is bounded by the simple closed, piece-wise smooth boundary curve S with positive orientation, and F is a vector field whose components have continuous first partial derivatives over some region containing S, then

Theorem 8.2: Irrotational Fields are Conservative

Suppose that F(x, y, z) is a vector field whose components have continuous first partial derivatives throughout the simply-connected open region D R3. Then, curl F = 0 in D if and only if for every simple closed curve C contained in the region D.

Theorem 8.3: Conservative Vector Fields

Conservative Vector Fields
If the vector field has components with continuous first partial derivatives in a simply-connected region on D, then the following five statements are equivalent (i.e., they are either all true or all false)

is conservative in D.
is a gradient field in D. (i.e., ).
is independent of path in D.
for every piecewise-smooth closed curve C lying in D.
is irrotational. (i.e., ).