Question

Using Stokes’ Theorem evaluate the line integral ∫C F .d l where F = yˆi + xz3 ˆj - zy3kˆ

and C is a circle x2 + y 2 = 4 in the plane z = −3.

 

 

 

25 Jan 2018
Answer :
Word Count : 610
To solve the given line integral using Stokes' Theorem, let's break it down: ### Stokes' Theorem Stokes' Theorem relates a surface integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary of the surface. The theorem states: \[ \int_C \mathbf{F} \cdot d\mathbf{l} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \] Where: - \( C \) is the closed curve (boundary of the surface \( S \)), - \( \mathbf{F} \) is the vector field, - \( \nabla \times \mathbf{F} \) is the curl of \( \mathbf{F} \), - \( d\mathbf{S} \) is the vector surface element. ### Given Problem The vector field is: \[ \mathbf{F} = y \hat{i} + xz^3 \hat{j} - zy^3 \hat{k} \] And the curve \( C \) is
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