Question

Prove (9.4) in the following way. Using (9.2) with, show that. Similarly, show thatand ∇. Letin that order form a right-handed triad (so that, etc.) and show that. Take the divergence of this equation and, using the vector identities (h) and (b) in the table at the end of Chapter 6, show that. The other parts of (9.4) are proved similar.

Short answer

Answer

The statement has been proven.

Step-by-step solution

Given Information

The value$u=x_1,x_3,x_3.$

Definition of Divergence.

The divergence is a scalar field produced by a vector operator that gives the quantity of a vector field's source at each location.

Prove the statement.

Evaluate for the value $u=x_1{x}_3,x_3$.

$$\begin{gathered} \nabla x_k=\sum _i\frac{1}{h_i}\frac{\partial x_k}{\partial x_i}{\stackrel{\wedge }{e}}_i \\ =\sum _i\frac{1}{h_i}\frac{1}{h_i}{\delta }_{ik} \\ {\stackrel{\wedge }{e}}_i\nabla x_i=\frac{1}{h_i}{\stackrel{\wedge }{e}}_i \end{gathered}$$

Evaluate the cross-product.

$$\begin{gathered} \nabla x_1\times \nabla x_2=\frac{1}{h_1{h}_2}{\stackrel{\wedge }{e}}_3 \\ \nabla x_i\times \nabla x_j=\frac{{\epsilon }_{ijk}}{h_i{h}_j}{\stackrel{\wedge }{e}}_3 \end{gathered}$$

Divergence of the equation is mentioned below.

$\nabla ·\left(\nabla x_i\times \nabla x_j\right)=\left(\nabla \times \nabla x_i\right)·\nabla x_j\times \nabla x_i·\left(\nabla \times \nabla x_j\right)$

Curl of the vector is always zero.

Equation becomes as follows.

$$\begin{gathered} \nabla ·\left(\nabla x_i\times \nabla x_j\right)=0 \\ \nabla ·\left(\frac{{\epsilon }_{ijk}}{h_i{h}_j}\stackrel{\wedge }{e}\right)=0 \end{gathered}$$

Hence, the statement has been proven.