Chapter 9: Vector Calculus
Section 9.4: Differential Identities
Example 9.4.11
For sufficiently well-behaved vector fields F=u i+v j+w k and G=a i+b j+c k, where u,v,w and a,b,c are functions of x,y,z, verify Identity 6 in Table 9.4.1.
Solution
Mathematical Solution
Identity 6: ∇·F×G=G·∇×F−F·∇×G
Its left side is the divergence of
F×G=ijkuvwabc=v c−w bw a−u cu b−v a
that is, it is ∂xv c−w b+∂yw a−u c+∂zu b−v a or
vxc+v cx−wxb−w bx+wya+w ay−uyc−u cy+uzb+u bz−vza−v az
or
awy−vz+buz−wx+cvx−uy−ucy−bz+vaz−cx+wbx−ay
Its right side is the difference of
abc·ijk∂x∂y∂zuvw = abc·wy−vzuz−wxvx−uy = awy−vz+buz−wx+cvx−uy
and
uvw·ijk∂x∂y∂zabc = uvw·cy−bzaz−cxbx−ay = ucy−bz+vaz−cx+wbx−ay
so the two sides match identically.
Maple Solution
Initialize
Tools≻Load Package: Student Vector Calculus
Loading Student:-VectorCalculus
Tools≻Tasks≻Browse: Calculus - Vector≻ Vector Algebra and Settings≻ Display Format for Vectors
Press the Access Settings button and select "Display as Column Vector"
Display Format for Vectors
Additional notational devices
The Suppress command in the Typesetting package allows suppression of arguments on input, as well as on output.
The declare command in the PDEtools package suppresses arguments on output, and sets partial derivatives as subscripts. Because the Suppress command acts first, the arguments can be suppressed in the ensuing declare command.
Typesetting:-Suppressux,y,z,vx,y,z,wx,y,z,ax,y,z,bx,y,z,cx,y,z
PDEtools:-declareu,v,w,a b c,quiet
Define the vector fields F and G
Write the free vector whose components are those of F. Context Panel: Evaluate and Display Inline
Context Panel: Student Vector Calculus≻Conversions≻To Vector Field
Context Panel: Assign to a Name≻F (or G, as appropriate)
u,v,w = →to Vector Field →assign to a nameF
a,b,c = →to Vector Field →assign to a nameG
Implement Identity 6: ∇·F×G=G·∇×F−F·∇×G
Common Symbols palette: Del, dot-product, and cross-product operators Press the Enter key.
Context Panel: Expand≻Expand
∇·F×G
vxc+vcx−wxb−wbx−uyc−ucy+wya+way+uzb+ubz−vza−vaz
G·∇×F−F·∇×G
awy−vz+buz−wx+cvx−uy−ucy−bz−vaz−cx−wbx−ay
= expand
Alternative demonstration of equality
Write the difference of the left-and right-hand sides and press the Enter key.
Context Panel: Simplify≻Simplify
∇·F×G−G·∇×F−F·∇×G
vxc+vcx−wxb−wbx−uyc−ucy+wya+way+uzb+ubz−vza−vaz−awy−vz−buz−wx−cvx−uy+ucy−bz+vaz−cx+wbx−ay
= simplify
0
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