Question:
Stokes theorem connects
Stokes theorem connects
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Stokes theorem converts a closed line integral into a surface integral of curl.
Updated On: Jul 3, 2026
- a line integral and a surface integral
- a line integral and a volume integral
- a surface integral and a volume integral
- gradient of a function and its surface integral
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The Correct Option is A
Solution and Explanation
Concept:
Stokes theorem states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of the vector field over a surface bounded by that curve. \[ \oint_C \vec F\cdot d\vec r = \iint_S (\nabla\times\vec F)\cdot \hat n\,ds \]
Step 1: Identify the left side. \[ \oint_C \vec F\cdot d\vec r \] is a line integral around the boundary curve \(C\).
Step 2: Identify the right side. \[ \iint_S (\nabla\times\vec F)\cdot \hat n\,ds \] is a surface integral over the surface \(S\).
Step 3: Final answer. Therefore, Stokes theorem connects \[ \boxed{\text{a line integral and a surface integral}} \]
Stokes theorem states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of the vector field over a surface bounded by that curve. \[ \oint_C \vec F\cdot d\vec r = \iint_S (\nabla\times\vec F)\cdot \hat n\,ds \]
Step 1: Identify the left side. \[ \oint_C \vec F\cdot d\vec r \] is a line integral around the boundary curve \(C\).
Step 2: Identify the right side. \[ \iint_S (\nabla\times\vec F)\cdot \hat n\,ds \] is a surface integral over the surface \(S\).
Step 3: Final answer. Therefore, Stokes theorem connects \[ \boxed{\text{a line integral and a surface integral}} \]
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