Q60P - Introduction to Electrodynamics

Question

Here are two cute checks of the fundamental theorems:

(a) Combine Corollary 2 to the gradient theorem with Stokes' theorem ( $v=\nabla T$, in this case). Show that the result is consistent with what you already knew about second derivatives.

(b) Combine Corollary 2 to Stokes' theorem with the divergence theorem. Show that the result is consistent with what you already knew.

Step-by-Step Solution

Answer

(a) Curl of gradient of function T is 0, that is $\nabla \times \nabla T=0$ .

(b) Divergence of curl of function v is 0, that is $\nabla \left(\nabla \times v\right)=0$ .

Step 1: Describe the given information

The curl of divergence and gradient of curl are to be proved equal to 0.

Step 2: Definestokes theorem and gradient theorem

The integral of curlof a function $f\left(x,y,z\right)$  over an open surface area is equal to the line integral of the function $\int \left(\nabla \times v\right)\cdot ds=\underset{l}{\oint }v\cdot dl$. According to the gradient theorem of corollary 2, the line integral of gradient of the function T  over a closed path is 0, that is, $\underset{l}{\oint }\nabla T\cdot dl=\mathbf{0}$

Step 3: Prove expression in part (a)

(a)

Stokes theorem is defined as $\int \left(\nabla \times v\right)\cdot da=\oint v\cdot dl$ . Substitute $\nabla T$  for v into $\int \left(\nabla \times v\right)\cdot da=\oint v\cdot dl$ as follows:

 $\int \left(\nabla \times \nabla T\right)\cdot da=\oint \nabla T\cdot dl$              ……….. (1)

 

Applying the gradient theorem of corollary 2, into equation (1), we obtain  

 $\begin{array}{d}\int \left(\nabla \times \nabla T\right)\cdot da=0\\ \nabla \times \nabla T=0\end{array}$

Thus, curl of gradient of function T is 0, that is, $\nabla \times \nabla T=0$ .

Step 4: Prove expression in part (b)

(b)

Applying the gradient theorem of corollary 2, into stokes theorem we obtain   

$\int \left(\nabla \times v\right)\cdot da=0$        …… (2)

 

The gauss divergence theorem states that the volume integral of the divergence of a function v is equal to the surface integral of the function v, that is, $\underset{v}{\int }\nabla v\cdot d\tau =\underset{s}{\int }vda$ 

Substitute $\nabla \times v$  for v into$\underset{v}{\int }\nabla v\cdot d\tau =\underset{s}{\int }vda$ .

$\underset{v}{\int }\nabla \left(\nabla \times v\right)\cdot d\tau =\underset{s}{\int }\left(\nabla \times v\right)da$    …… (3)

 

Comparing equations (2) and (3).

 $\begin{array}{c}\underset{v}{\int }\nabla \left(\nabla \times v\right)\cdot d\tau =0\\ \nabla \left(\nabla \times v\right)=0\end{array}$

 

Thus, divergence of curl of function v is 0, that is $\nabla \left(\nabla \times v\right)=0$ .