Stoke's Theorem

Q. If a vector field $\overrightarrow{V}$ is related to another field $\overline{A}$ through $\overrightarrow{V}$ = $\nabla \times \overline{A}$, which of the following is true?

Note: C and $S_c$ refer to any closed contour and any surface whose boundary is C.

1. ${\oint }_c\overrightarrow{V}.dl={\iint }_{S_c}\overrightarrow{A}.\overrightarrow{d}s$
2. ${\oint }_c\overrightarrow{A}.dl={\iint }_{S_c}\overrightarrow{V}.\overrightarrow{d}s$
3. ${\oint }_c\nabla .\overrightarrow{V}.d\overrightarrow{l}={\iint }_{S_c}\nabla .\overrightarrow{A}.\overrightarrow{d}s$
4. ${\oint }_c\nabla .\overrightarrow{A}.d\overrightarrow{l}={\iint }_{S_c}\overrightarrow{V}.\overrightarrow{d}s$

Q. Stokes theorem connects

1. a line integral and a surface integral
2. a surface integral and a volume integral
3. a line integral and a volume integral
4. gradient of a function and its surface integral

Q. $\int \int (\nabla \times P)$.ds, where P is a vector , is equal to

1. $\oint P.dl$
2. $\oint \nabla \times \nabla \times P.dl$
3. $\oint \nabla \times P.dl$
4. $\iiint \nabla .Pdv$

Q. Consider a vector field $\overrightarrow{A}\left(\overrightarrow{r}\right)$. The closed loop line integral $\oint \overrightarrow{A}.d\overrightarrow{l}$ can be expressed as

1. $∯$$(\nabla \times \overrightarrow{A}).\overrightarrow{d}s$ over the closed surface bounded by the loop
2. $∯$$(\nabla \times \overrightarrow{A})dv$ over the closed volume bounded by the loop
3. $\iiint (\nabla .\overrightarrow{A})dv$ over the open volume bounded by the loop
4. $\iint (\nabla \times \overrightarrow{A}).\overrightarrow{d}s$ over the open surface bounded by the loop

Q. $\int \int (\nabla \times P).dS,$ Where P is a vector, is equal to

1. $\oint P.dl$
2. $\oint \nabla \times \nabla \times P.dl$
3. $\oint \nabla \times P.dl$
4. $\int \int \int \nabla .Pdv$