Vector Calculus | Part IA, 2006

State Stokes' theorem for a vector field A\mathbf{A}.

By applying Stokes' theorem to the vector field A=ϕk\mathbf{A}=\phi \mathbf{k}, where k\mathbf{k} is an arbitrary constant vector in R3\mathbb{R}^{3} and ϕ\phi is a scalar field defined on a surface SS bounded by a curve S\partial S, show that

SdS×ϕ=Sϕdx\int_{S} d \mathbf{S} \times \nabla \phi=\int_{\partial S} \phi d \mathbf{x}

For the vector field A=x2y4(1,1,1)\mathbf{A}=x^{2} y^{4}(1,1,1) in Cartesian coordinates, evaluate the line integral

I=AdxI=\int \mathbf{A} \cdot d \mathbf{x}

around the boundary of the quadrant of the unit circle lying between the xx - and yy axes, that is, along the straight line from (0,0,0)(0,0,0) to (1,0,0)(1,0,0), then the circular arc x2+y2=1,z=0x^{2}+y^{2}=1, z=0 from (1,0,0)(1,0,0) to (0,1,0)(0,1,0) and finally the straight line from (0,1,0)(0,1,0) back to (0,0,0)(0,0,0).

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