Chapter 18: Line Integrals and Surface Integrals

Section 18.1 Line Integrals Section 18.7 Surface Integrals a. Work Done by Varying Force Over a Curved Path a. Mass of a Material Surface b. Definition b. Surface Integrals c. Theorem 18.1.4 c. Average Value of H on S d. Piecewise Smoothness d. Moment of Inertia e. Properties of Piecewise Smooth Curves e. Flux of a Vector Field

Section 18.2 Fundamental Theorem for Line Integrals Section 18.8 The Vector Differential Operator ∇ a. Theorem 18.2.1 a. Vector Differential Operator b. Corollary and Example b. Divergence c. Curl Section 18.3 Work-Energy Formula; Conservation of Mechanical Energy d. Theorems for Divergence and Curl a. Work-Energy Formula e. Identities and Properties b. Conservative Field, Potential Energy Functions f. The Laplacian c. Conservation of Mechanical Energy Section 18.9 The Divergence Theorem Section 18.4 Another Notation for Line Integrals; Line Integrals with a. Theorem 18.9.2 Respect to Arc Length b. Divergence as Outward Flux per Unit Volume a. Line Integral with Respect to Arc Length c. Solids Bounded by Two or More Closed Surfaces b. Properties for a Thin Wire Section 18.10 Stokes’s Theorem Section 18.5 Green’s Theorem a. Theorem 18.10.1, Stokes’s Theorem a. Green’s Theorem, Theorem 18.5.1 b. Partial Converse to Stokes Theorem b. Type I and Type II Region c. Irrotational Flow c. Area of a Jordan Region d. Theorem 18.10.3 d. Regions Bounded by Two or More Jordan Curves

Section 18.6 Parametrized Surfaces; Surface Area a. Parametrized Surfaces b. Fundamental Vector Product - Illustration c. Fundamental Vector Product - Notation d. Area of a Parametrized Surface e. Area of a Smooth Surface Salas, Hille, Etgen Calculus: One and Several Variables f. Area of Surface z = f (x,y) Main Menu Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Line Integrals

The Work Done by a Varying Force over a Curved Path The work done by a constant force F on an object that moves along a straight line is, by definition, the component of F in the direction of the displacement multiplied by the length of the displacement vector r (Project 13.3):

W = (compd F)||r||.

Such a curve is said to be piecewise smooth.

If a force F is continually applied to an object that moves over a piecewise- smooth curve C, then the work done by F is the line integral of F over C:

Example Evaluate the line integral h( r) ⋅ dr ∫C where C is the square with vertices (0, 0), (1, 0), (1, 1), (0, 1) traversed counterclockwise and h (x, y) = (3x2 y + xy2 − 1) i + (x3 + x2 y + 4y3) j.

Solution First we try to determine whether h is a gradient. The functions P(x, y) = 3x2 y + xy2 − 1 and Q(x, y) = x3 + x2 y + 4y3 are continuously differentiable everywhere, and ∂∂PQ =+=32x2 xy ∂∂yx

Therefore h is the gradient of a function f . By (18.2.2), h( r) ⋅=∇ drf ( r) ⋅= dr 0 ∫∫CC

This relation is called the work–energy formula.

Conservative Force Fields In general, if an object moves from one point to another, the work done (and hence the change in kinetic energy) depends on the path of the motion. There is, however, an important exception: if the force field is a gradient field, F = ∇f then the work done (and hence the change in kinetic energy) depends only on the endpoints of the path, not on the path itself. (This follows directly from the fundamental theorem for line integrals.) A force field that is a gradient field is called a conservative field. Since the line integral over a closed path is zero, the work done by a conservative field over a closed path is always zero. An object that passes through a given point with a certain kinetic energy returns to that same point with exactly the same kinetic energy.

Potential Energy Functions Suppose that F is a conservative force field. It is then a gradient field. Then −F is also a gradient field. The functions U for which ∇=−U F are called potential energy functions for F.

The Conservation of Mechanical Energy

As an object moves in a conservative force field, its kinetic energy can vary and its potential energy can vary, but the sum of these two quantities remains constant. We call this constant the total mechanical energy.

The total mechanical energy is usually denoted by the letter E. The law of conservation of mechanical energy can then be written

Suppose that f is a scalar field continuous on a piecewise-smooth curve C : r (u) = x(u) i + y(u) j + z(u) k, u ∈ [a, b].

If s(u) is the length of the curve from the tip of r (a) to the tip of r (u), then, as you have seen,

2 22 su′′( ) =r ( u) =  xu ′( )  + yu ′( )  + zu ′( ) 

The line integral of f over C with respect to arc length s is defined by setting

Suppose now that C represents a thin wire (a material curve) of varying mass density λ = λ(r). (Here mass density is mass per unit length.) The length of the wire can be written

The mass of the wire is given by

and the center of mass rM can be obtained from the vector equation

The moment of inertia about an axis is given by the formula

Regions Bounded by Two or More Jordan Curves

is called the fundamental vector product of the surface.

The Area of a Parametrized Surface

More generally, let’s suppose that we have a surface S parametrized by a continuously differentiable function r = r (u, v), (u, v) ∈ Ω. We assume that Ω is a basic region in the uv-plane and that r is one-to-one on the interior of Ω. (We don’t want r to cover parts of S more than once.) Also we assume that the fundamental vector product N = r´u × r´v is never zero on the interior of Ω. (We can then use it as a normal.) Under these conditions we call S a smooth surface and define

The Area of a Surface z = f (x, y) Above each point (x, y) of Ω there is one and only one point of S. The surface S is then the graph of a function z = f (x, y), (x, y) ∈ Ω .

As we show, if f is continuously differentiable, then

The Mass of a Material Surface Imagine a thin distribution of matter spread out over a surface S. We call this a material surface.

Surface Integrals The double integral in (18.7.1) can be calculated not only for a mass density function λ but for any scalar field H continuous over S. We call this integral the surface integral of H over S and write

Note that, if H(x, y, z) is identically 1, then the right-hand side of (18.7.2) gives the area of S. Thus

Like the other integrals you have studied, the surface integral satisfies a mean- value condition; namely, if the scalar field H is continuous, then there is a point (x0, y0, z0) on S for which σ = ∫∫ Hxyzd( ,, ) Hx( 0, yz 00 ,)( area of S) S

We call H(x0, y0, z0) the average value of H on S. Thus we can write

We can also take weighted averages: if H and G are continuous on S and G is nonnegative on S, then there is a point (x0, y0, z0) on S for which

As you would expect, we call H(x0, y0, z0) the G-weighted average of H on S.

Suppose that a material surface S rotates about an axis. The moment of inertia of the surface about that axis is given by the formula

where λ = λ(x, y, z) is the mass density function and R(x, y, z) is the distance from the axis to the point (x, y, z). (As usual, the moments of inertia about the x, y, z axes are denoted by Ix , Iy , Iz .)

The Flux of a Vector Field Suppose that S : r = r (u, v), (u, v) ∈ Ω is a smooth surface with a unit normal n = n(x, y, z) that is continuous on all of S. Such a surface is called an oriented surface. Note that an oriented surface has two sides: the side with normal n and the side with normal −n. If v = v(x, y, z) is a vector field continuous on S, then we can form the surface integral

This surface integral is called the flux of v across S in the direction of n.

The vector differential operator  is defined formally by setting

If v = v1 i + v2 j + v3 k is a differentiable vector field, then, by definition,

The “product,”  · v, defined in imitation of the ordinary dot product, is called the divergence of v:  · v = div v.

The “product,”  × v, defined in imitation of the ordinary cross product, is called the curl of v:  × v = curl v.

The next two identities are product rules. Here f is a scalar field and v is a vector field.

As usual, set r = x i + y j + z k and r = r .We know that  · r = 3 and × r = 0 at all points of space. Now we can show that if n is an integer, then, for all r ≠ 0,

The Laplacian From the operator ∇ we can construct other operators, the most important of which is the Laplacian ∇2 =∇⋅∇. The Laplacian (named after the French mathematician Pierre-Simon Laplace) operates on scalar fields according to the following rule:

Divergence as Outward Flux per Unit Volume Choose a point P and surround it by a closed ball N∈ of radius ∈. According to the divergence theorem,

∇⋅ = ∫∫∫ ( vv)dx dy dz flux of out of N∈ N∈

Think of v as the velocity of a fluid. As suggested in Section 18.8, negative divergence at P signals an accumulation of fluid near P:

∇⋅v <0 at PN ⇒ flux out of ∈∈< 0 ⇒ net flow into N .

Positive divergence at P signals a flow of liquid away from P:

∇⋅v>0 at PN ⇒ flux out of ∈∈> 0 ⇒ net flow out of N . Points at which the divergence is negative are called sinks; points at which the divergence is positive are called sources. If the divergence of v is 0 throughout, then the flow has no sinks and no sources and v is called solenoidal.

Solids Bounded by Two or More Closed Surfaces ∫∫∫ (∇⋅v)dx dy dz =∫∫ (vn ⋅ )dσ TT11bdry of

∫∫∫ (∇⋅v)dx dy dz =∫∫ (vn ⋅ )dσ TT22bdry of

∫∫∫(∇⋅v)dx dy dx = ∫∫ (vn ⋅ ) dσ TS

The Normal Component of ∇× v as Circulation per Unit Area; Irrotational Flow

Take a point P within the flow and choose a unit vector n. Let D∈ be the ∈-disk that is centered at P and is perpendicular to n. Let C∈ be the circular boundary of D∈ directed in the positive sense with respect to n. By Stokes’s theorem,

∫∫( ∇×n) ⋅nd σ = ∫ v( r) ⋅ dr C∈ D∈

At each point P the component of ∇× v in any direction n is the circulation of v per unit area in the plane normal to n. If ∇× v = 0 identically, the fluid has no rotational tendency, and the flow is called irrotational.