confirm that F(x,y)=(X+y,X-y) is the gradient of f(X,y)=(1/2)X ^2+xy-(1/2)y^2. T
ID: 2854222 • Letter: C
Question
confirm that F(x,y)=(X+y,X-y) is the gradient of f(X,y)=(1/2)X ^2+xy-(1/2)y^2. Then use the fundamental theorem for line integral a to evaluate F(X).dx from (0,0) to (1,1). confirm that F(x,y)=(X+y,X-y) is the gradient of f(X,y)=(1/2)X ^2+xy-(1/2)y^2. Then use the fundamental theorem for line integral a to evaluate F(X).dx from (0,0) to (1,1). confirm that F(x,y)=(X+y,X-y) is the gradient of f(X,y)=(1/2)X ^2+xy-(1/2)y^2. Then use the fundamental theorem for line integral a to evaluate F(X).dx from (0,0) to (1,1).Explanation / Answer
f(x,y) = (1/2)x2 + xy - (1/2)y2
gradient ==> f(x,y) = <fx , fy>
= <(1/2)(2x2-1) + (1)y - 0 , 0 + x(1) - (1/2)(2y2-1)>
= <x+y , x-y>
Line integal c F.dr = f(1,1) - f(0,0)
==> [(1/2)(1)2 + (1)(1) - (1/2)(1)2] - [(1/2)(0)2 + (0)(0) - (1/2)(0)2]
==> [1/2 + 1 - 1/2] - [0 + 0 - 0]
==> 1
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