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Q. 13

Found in: Page 1140


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Why is dA in Green’s Theorem replaced by dS in Stokes’ Theorem?

dArefers to ordinary area while dSrefers to the surface area. So,dA in Green's Theorem is replaced by dS in Stokes' Theorem because .

See the step by step solution

Step by Step Solution

Step 1. Given Information

dA in Green’s Theorem replaced by dS in Stokes’ Theorem.

Step 2. Understanding both the Theorems

Green's Theorem states that;"Let R be a region in the plane with smooth boundary curve C oriented counterclockwise by r(t)=(x(t),y(t)) for atb.If a vector field F(x,y)=F1(x,y),F2(x,y) is defined by R, then, CF·dr=RF2x-F1ydA."Stokes' Theorem states that;"Let be S an oriented, smooth or piecewise-smooth surface bounded by a curve C. Suppose that n is an oriented unit normal vector of S and C has a parametrization that traverses C in the counterclockwise direction with respect to n.If a vector field F(x,y,z)=F1(x,y,z)i+F2(x,y,z)j+F3(x,y,z)kis defined on S, then, CF(x,y,z)·dr=ScurlF(x,y,z)·ndS".

Step 3. Understanding the notations

Both the notations dA and dS refer to area.In the case of Green's Theorem, dA refers to the ordinary area.In the case of Stokes' Theorem, dS refers to the surface area. In this case, dS keeps track of any distortion to the area of the surface S caused by transforming a patch of the uv-plane to parametrize S. Hence, dA in Green's Theorem can be replaced by dS in Stokes' Theorem.

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