Chapter 9: Vector Calculus
Section 9.3: Differential Operators
A full vector calculus course would discuss the five differential operators: gradient, divergence, curl, Laplacian, and D'Alembertian. The first four of these, listed in Table 9.3.1, are appropriate for an introduction to vector calculus. Table 9.3.1 gives a brief physical interpretation of each of these operators.
Measure of change in the scalar function f
Local measure of "spread" of the vector field F
Local measure of "rotation" in the vector field F
The divergence (i.e., spread) of the gradient field of the scalar f
Table 9.3.1 Interpreting the four basic differential operators of vector calculus
The divergence of a field F is best understood after the notion of the flux of a field through a boundary, either a curve or a surface, is mastered. But flux is defined via integrals of vector fields, something that will be studied in Section 9.5 (line integrals) and Section 9.6 (surface integrals). Loosely speaking, however, flux is a measure of the "flow" of the field F through a boundary. Divergence is the limiting ratio of the flux (through a closed boundary) to the area or volume enclosed by that boundary; hence, a "local" measure of spread.
The curl at a point is a measure of the local "rotation" in the field, determined as the limiting ratio of the surface integral of N×F and the enclosed volume. (The vector N×F is tangent to the surface for which N is the unit normal.)
Examples illustrating the meanings of both divergence and curl are postponed until after the appropriate integration topics have been studied.
For different coordinate systems, Table 9.3.2 lists the expressions for the gradient and Laplacian of the scalar f, and the divergence of the vector F, whose components are u,v, and w. The fonts are reduced so the expressions fit into the table. Except for basis vectors, subscripts are used to represent partial derivatives.
Laplacian: ∇2f= ∇·∇f
fx i+fy j+fz k
fr er+fθ/rer+fz k
fρ eρ+fφρ eφ+fθρ sinφ eθ
ρ2uρρ2+v sinφφρ sinφ+wθρ sinφ
Table 9.3.2 Gradient, divergence, and Laplacian in different coordinate systems
For different coordinate systems, Table 9.3.3 details the curl of the vector field F, whose components are u,v, and w. In each case, a mnemonic (memory device) is given on the left, and the resulting vector ∇×F is given on the right. The mnemonic is a determinant that is expanded by its top row.
erreθkr∂r∂θ∂zur vw=wθ−(r v)zruz−wr(r v)r−uθr
eρρ2sinφeφρ sinφeθρ∂ρ∂φ∂θuρ vρ w sinθ=w cos(φ)ρ sin(φ)+wφρ−vθρ sin(φ)uθρ sin(φ)−wρ−wρvρ+vρ−uφρ
Table 9.3.3 ∇×F in different coordinate systems
Each of the nonCartesian expressions in Tables 9.3.(2-3) can be obtained by mapping the Cartesian equivalent to the new coordinate system. Nine of the seventeen examples below involve computations of this type.
For fx,y=2 x2+3 y2, obtain ∇f at the point x,y=1,1. Show that at this point, the gradient vector is orthogonal to a vector tangent to the level curve through this point.
If r and θ are polar coordinates, obtain ∇f for fr,θ=lnr tanθ.
Derive the expression for the gradient in polar coordinates.
Integrate the gradient field for the scalar function fx,y=x y.
Obtain the divergence of the Cartesian vector field F=x y i+x/y j.
Change the Cartesian vector field F=x y i+x/y j to polar coordinates, and obtain its divergence in those coordinates. Then express the result in Cartesian coordinates and compare to the result in Example 9.3.5.
Derive the expression for the divergence in polar coordinates.
Compute the curl of the Cartesian vector field F=x y i−y z j−x z k.
Change the Cartesian vector field F=x y i−y z j−x z k to cylindrical coordinates and obtain its curl in those coordinates. Then express the result in Cartesian coordinates and compare to the result in Example 9.3.8.
Derive the expression for the curl in cylindrical coordinates.
Compute the curl of the Cartesian vector field F=y i−z j−x k, then change to spherical coordinates and again obtain the curl. Transform the result back to Cartesian coordinates and compare to the original results.
Derive the expression for the curl in spherical coordinates.
Obtain the Laplacian of the scalar function fx,y=x/y+y/x. Show that it is equivalent to the divergence of the gradient.
Change fx,y=x/y+y/x to polar coordinates, and in those coordinates obtain the Laplacian. Then show the results are equivalent to those in Example 9.3.13.
Derive the expression for the Laplacian in polar coordinates.
Graph the vector fields F1=x i+y j and F2=F1/x2+y2. Show that ∇·F1=2, but ∇·F2=0, even though the arrows of both fields are radially outward, suggesting "divergence" for both.
Graph the vector fields F1=y i−x j and F2=F1/x2+y2. Show that ∇×F1=−2 k, but ∇×F2=0, even though the arrows of both fields are tangent to concentric circles, suggesting "rotation" for both.
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