Merced College; Don Power

MULTIVARIATE CALCULUS - CH 16, LECTURE

TOPICS IN VECTOR CALCULUS

16.1 Vector Fields

Sample Maple commands for fieldplots:

with(plots):

f:=[sin(x),1];

fieldplot(f,x=-10..10,y=-10..10);

fieldplot3d([2*x,2*y,1],x=-1..1,y=-1..1,z=-1..1,grid=[4,4,4]);

Inverse-square fields: definition

Del operator: =

Gradient field: F(x,y,z) = Scalar --> Vector

That is, F = del(f) for some function f

Then we say the function f is a potential function for the gradient field F

And we say that the vector field is conservative.

Important fact: Inverse square fields are conservative, as long as the region does not include the origin

If F = <f,g,h> is a vector field,

div F = f_x + f_y + f_z = (is there a net flow out of or into a region within a flow field?)

Vector --> Scalar

curl F = (which way would a paddlewheel turn in a flow field?)

Vector --> Vector

16.2 Line Integrals

Interpretations:

Area of a curtain over a curved path

Mass of a curved wire

Sugar spilled on a table, picked up by a child's moistened finger

Scalar form: , same as

Vector form: , same as

16.3 Independence of Path; Conservative Vector Fields

Analog with elementary integrals of one variable: int(f,x,a,b)= F(b)-F(a) if antideriv F exists.

If F is conservative, then φ exists such that grad(φ)=F;

Then int(F dot dr) on a curve C = φ(x₁,y₁,z₁)-φ(x₀,y₀,z₀)

This is the fundamental theorem of work integrals ("fundamental theorem of line integrals" in most texts, though it applies only to conservative fields)

We say integral is "independent of path" (Why?)

Test to determine whether a field is conservative: f_y = g_x, or curl F = 0

Recovering the potential function φ from the vector field F

Examples X13, 12

Useful exercise: For X11, evaluate as ordinary line integral

1. along path y = 2/3 x

2. along path y = 2/9 x^2

3. using fundamental theorem of work integrals

16.4 Green's Theorem

Second simplification of line integrals:

Now, field need not be conservative

Applies to line integrals in a simple, closed, piecewise smooth curve in a plane

Notation (integral symbol) for a closed curve: or just

Green's Thm: Line int (f dx + g dy) = double int ( g_x - f_y) dA,

where orientation of curve C is CCW

Examples

Areas

Three formulas, each of which, by Green's Thm, gives

or or

X24: show that by applying Green's Thm to get

Multiply connected regions

Take sum over each curve

Outer curve is oriented CCW

Curve around each curve is oriented CW

Note that result for a conservative field would be 0, since g_x - f_y = 0 in a conservative field

16.5 Surface Integrals

Thm 1: Surface integral for f(x,y,z) over a surface given by r = <x(u,v),y(u,v),z(u,v)>

Ex: X28

Thm 2: Surface integral for f(x,y,z) over a surface given by a function z =g(x,y), y =g(x,z), or x =g(y,z)

Ex: X6

16.6 Applications of Surface Integrals; Flux

Orientation of a surface:

Positive orientation: n is the unit vector crossprod(r_u,r_v) / norm(crossprod(r_u,r_v)

Negative orientation: orientation determined by -n

Upward orientation: z-component of the normal vector is positive

For z = f(x,y), let G(x,y,z) = z - f(x,y) and take grad(G)

Downward orientation: z-component of the normal vector is positive

For z = f(x,y), let G(x,y,z) = f(x,y) - z and take grad(G)

Ex: X6: Find unit normal that defines positive orientation; is the orientation inward or outward? Is it upward or downward?

Flux = doubleint ( F dot n dS) where F represents the velocity field of an incompressible fluid

= doubleint (F dot (crossprod(r_u,r_v) dA) for surface r = <x(u,v),y(u,v),z(u,v)>

=doubleint (F dot grad(G) dA) for surface G(x,y,z)=0,

where G = z-f(x,y) (for upward unit normal) or f(x,y)-z (for downward unit normal) etc.

Examples

X1 for one face (say y=1) by first formula (definition)

X13, for a surface given parametrically or as a vector function

X9, for a surface given as an equation in x,y, and z

For G, we can either:

solve for z first, set = 0, then find grad(G), or

set = 0, find the grad(result), then divide through as necessary to find grad(G)

16.7 The Divergence Theorem (Gauss' Theorem)

Let G be a solid in space whose surface is oriented outward, let F be a force field in space (with continuous first partials on an open set containing G), let n be the outward unit normal for the surface:

then

Ex 1 in text: sphere. Since div F is a constant, we do not even need to integrate.

Ex: X18 from 16.6. Since div F is 0, we do not even have to set up integral.

This is usually the best method to find the flux across a closed surface, in particular when the surface consists of multiple sections, as with a cube, a cylindrical solid, or a hemispherical solid

Ex of cube: Do X17a from 16.6 by divergence theorem

Ex of hemispherical solid: Do X12: div F = 5, result is 5pa⁴/4

Ex of cylindrical solid: X9: div F = 3x²+3y²+3z²; result is 180p.

Ex where region is tougher to set up: X15: For solid bounded by z=4-x², y+z=5, z=0 and y=0,

use 0£y£5-z, then 0£z £4-x², then -2£x£2.

Interpretation: pg 1150 and 1151

Gauss' Law for inverse square fields and application to electrostatics: See text 1151-2

16.8 Stokes' Theorem

Situation: An oriented surface in space is bounded by a simple curve. If the curve C is oriented positively relative to the direction of the unit normal vectors to the surface, then

Examples: Work (Ex 1)

Ex 2: verifies the formula -- calculate the result both ways

Note that if the surface is in or parallel to the xy-plane, Stokes' Thm reduces to Green's Thm

Physical interpretation: circulation of a fluid

For an intuitive picture, see the paddlewheel diagram on page 1158

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