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GATE ECE 2009 | Question-23

If a vector field $\vec{V}$ is related to another vector field $\vec{A}$ through $\vec{V}=\nabla \times \vec{A}$, which of the following is true ? Note: $C$ and $S_{C}$ refer to any closed contour and any surface whose boundary is $C$.

  1. $\oint_{C} \vec{V} \cdot \overrightarrow{d l}=\int \int_{S_{C}} \vec{A} \cdot \overrightarrow{d S}$
  2. $\oint_{C} \vec{A} \cdot \overrightarrow{d l}=\int \int_{S_{C}} \vec{V} \cdot \overrightarrow{d S}$
  3. $\oint_{C} \nabla \times \vec{V} \cdot \overrightarrow{d l}=\int \int_{S_{C}} \nabla \times \vec{A} \cdot \overrightarrow{d S}$
  4. $\oint_{C} \nabla \times \vec{A} \cdot \overrightarrow{d l}=\int \int_{S_{C}} \vec{V} \cdot \overrightarrow{d S}$
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