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Let g be a function and F be a vector field, and assume that g and F are defined on a simple region R in the ry plane with a piecewise smooth boundary C oriented counterclockwise. Assume that the partial derivatives of g and the components of F exist and are continuous throughout R. 1. Using the identity V. (GF) = 9V · F + Vg · F and the form of Green's Theorem F.nds = V.FdA R show that gF -nds = || i9v -F + Vg · F]dA 2. Using (1), let f be a function of two variables having continuous partial derivatives throughout R. Show that R Hint: Let F = Vf in problem 1. Remember the Laplacian is V²f = V · Vƒ

Let g be a function and F be a vector field, and assume that g and F are defined on a simple region R in the ry plane with a piecewise smooth boundary C oriented counterclockwise. Assume that the partial derivatives of g and the components of F exist and are continuous throughout R. 1. Using the identity V. (GF) = 9V · F + Vg · F and the form of Green's Theorem F.nds = V.FdA R show that gF -nds = || i9v -F + Vg · F]dA 2. Using (1), let f be a function of two variables having continuous partial derivatives throughout R. Show that R Hint: Let F = Vf in problem 1. Remember the Laplacian is V²f = V · Vƒ

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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Let g be a function and F be a vector field, and assume that g and F are defined on a simple region R in the ry plane with a piecewise smooth boundary C oriented counterclockwise. Assume that the partial derivatives of g and the components of F exist and are continuous throughout R. 1. Using the identity V. (GF) = gV · F + Vg · F and the form of Green's Theorem F.nds = V.FdA R show that gF.n ds = [9V · F + Vg · F] dA 2. Using (1), let f be a function of two variables having continuous partial derivatives throughout R. Show that nds = [/ov²s+* Vg. Vf)dA R Hint: Let F = Vf in problem 1. Remember the Laplacian is V²f = V Vf
Transcribed Image Text:Let g be a function and F be a vector field, and assume that g and F are defined on a simple region R in the ry plane with a piecewise smooth boundary C oriented counterclockwise. Assume that the partial derivatives of g and the components of F exist and are continuous throughout R. 1. Using the identity V. (GF) = gV · F + Vg · F and the form of Green's Theorem F.nds = V.FdA R show that gF.n ds = [9V · F + Vg · F] dA 2. Using (1), let f be a function of two variables having continuous partial derivatives throughout R. Show that nds = [/ov²s+* Vg. Vf)dA R Hint: Let F = Vf in problem 1. Remember the Laplacian is V²f = V Vf
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