Please check my work for the following problem:(adsbygoogle = window.adsbygoogle || []).push({});

The problem statement, all variables and given/known data

By subsitutingA(r) =c[tex]\phi[/tex](r) in Gauss's and Stokes theorems, wherecis an arbitrary constant vector, find these two other "fundamental theorems":

a) [tex] \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds[/tex]

b) [tex]- \int_{S} \nabla \phi \times ds = \int_{C} \phi dl[/tex]

The attempt at a solution

So I start with 'a' and I'll subsitute:A(r) =c[tex]\phi[/tex](r)

Original equation:

[tex] \int_{\tau} (\nabla \cdot A) d \tau = \int_{S} A \cdot ds[/tex]

Subsitution:

[tex] \int_{\tau} (\nabla \cdot c \phi) d \tau = \int_{S} c \phi \cdot ds[/tex]

[tex]c \int_{\tau} (\nabla \cdot \phi) d \tau = c \int_{S} \phi \cdot ds[/tex]

this leads us back to the equation that we want:

[tex] \int_{\tau} \nabla \phi d \tau = \int_{S} \phi ds[/tex]

right?

So I start with 'b' and I'll subsitute:A(r) =c[tex]\phi[/tex](r)

Original equation:

[tex] \int_{s} (\nabla \times A) \cdot ds = \int_{C} A \cdot dl[/tex]

Subsitution:

[tex] \int_{s} (\nabla \times c \phi) \cdot ds = \int_{C} c \phi \cdot dl[/tex]

[tex] \int_{s} \nabla \cdot ( c \phi \times ds) = c \int_{C} \phi \cdot dl[/tex]

i am stuck here on what to do for part b.

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# Homework Help: Gauss' and Stokes' Theorems

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