MATH503 hw2 - (b) R C ( f g ) t ds = RR S ( f g ) n dS ....

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Homeworks (Math 503) T1 : Div, grad, curl and all that , by Schey (4th edition) T2 : Calculus of variations T3 : Nonlinear dynamics and chaos , by Strogatz Note: Detail your work to receive full credit Homework#2: (due Wed Oct 6) 1. Evaluate RR S ( ∇ × F ) · n dS where F = yz i - xz j + xz k , and S is the triangle with vertices (1 , 2 , 8), (3 , 1 , 9), (2 , 1 , 7) with upward normal. 2. Evaluate R C F · t ds where F = sin( x ) i +cos( y ) j + xz k , and C is given by r ( u ) = u 3 i - u 2 j + u k for 0 u 2 π . 3. Verify Stokes’ theorem for F = ( x + y 2 ) i + ( y + z 2 ) j + ( z + x 2 ) k where C is the triangle with vertices (1 , 0 , 0), (0 , 1 , 0), (0 , 0 , 1) oriented counterclockwise. 4. Evaluate R C ( y + sin x ) dx + ( z 2 + cos y ) dy + x 3 dz where C is the curve r ( θ ) = cos( θ ) i + sin( θ ) j + sin(2 θ ) k for 0 θ 2 π . Hint: Observe that C lies on the surface z = 2 xy and use Stokes’ theorem. 5. (a) Show that ∇ × ( f F ) = f ∇ × F + ( f ) × F . Using Stokes’ theorem, show that:
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Unformatted text preview: (b) R C ( f g ) t ds = RR S ( f g ) n dS . (c) R C ( f g-g f ) t ds = 0. 6. In the following cases, show that F is conservative. Then, nd the solution f such that F = f : (a) F = sin( y ) i + x cos( y ) j-sin( z ) k . (b) F = (1 + xy ) e xy i + ( e y + x 2 e xy ) j . 7. In analogy with a uid mechanics example given in class, use Greens theorem with a suitable choice of the velocity eld u ( ) to show that, if the stream function satises = 0 in a region S , then ZZ S h x 2 + y 2 i dxdy = Z C N du ....
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