Faraday's Law and Stokes Theorem

[CGI]

Homework Statement

Homework Equations

Stokes Theorem

The Attempt at a Solution

I'm having a tough time "cancelling" out integrals from both sides of an equation (if possible). On the right hand of the equation, we know since it is a closed curve, that Stoke's Theorem applies and we can change the integral of ∫E⋅dr to ∫∇×E dA. On the left hand of the equation, we can insert d/dt into the integrand since it is independent of dA, giving us ∫dB/dt dA.

Overall, After moving the negative side over, we have:
-∫dB/dt dA = ∫∇×E dA

Now, from what my TA told me, he said there was a way to justify the "cancellation" of the integrals. I'm assuming they have the same bounds, but he gave me a hint and said it was something related to what I learned in Calc I, but having done so in only 1D.

Anyone have any ideas? It would be much appreciated!

 

[Ray Vickson]

Homework Statement

View attachment 220817

Homework Equations

Stokes Theorem

The Attempt at a Solution

I'm having a tough time "cancelling" out integrals from both sides of an equation (if possible). On the right hand of the equation, we know since it is a closed curve, that Stoke's Theorem applies and we can change the integral of ∫E⋅dr to ∫∇×E dA. On the left hand of the equation, we can insert d/dt into the integrand since it is independent of dA, giving us ∫dB/dt dA.

Overall, After moving the negative side over, we have:
-∫dB/dt dA = ∫∇×E dA

Now, from what my TA told me, he said there was a way to justify the "cancellation" of the integrals. I'm assuming they have the same bounds, but he gave me a hint and said it was something related to what I learned in Calc I, but having done so in only 1D.

Anyone have any ideas? It would be much appreciated!

Basically, you have that $\int \int_S \mathbf{U} \cdot d\mathbf{A}= \int \int_S \mathbf{V} \cdot d\mathbf{A}$ for every "suitable" region $S$, and then need to show that this gives $\mathbf{U} = \mathbf{V}$ at all points.

The standard way to try proving such things is to apply it so an "infinitesimal" region $A$ surrounding a point $\mathbf{r}_0 = (x_0,y_0,z_0)$ and appeal to continuity of the vector fields $\mathbf{U}$ and $\mathbf{V}$ around $\mathbf{r}_o$ to show that we must have $\mathbf{U}(\mathbf{r}_0) = \mathbf{V}(\mathbf{r}_0)$.