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M 427L - Practice Final 1

# M 427L - Practice Final 1 - M 427L Practice for the third...

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M 427L Practice for the third exam. The actual exam will not be so long 1. Let vector T ( r, θ ) : R 2 R 3 be a differentiable function. Which of the following is the geometric meaning of ( vector T ∂r × vector T ∂θ ) · dr · (a) This is a vector normal to the surface, with length (roughly) the area of the image of a tiny rectangle of area dr · . (b) This is the axis of (infintesimal) rotation for flow of the vector field vector T , its length is (roughly) the angular velocity. (c) This is a vector which points in the direction of maximal increase of vector T (at the point ( r, θ )). (d) This is the determinant of the Jacobian matrix for vector T , and so gives the factor by which vector T stretches (infintesimally). For 2 –3 let vector T : R 2 R 3 be the function vector T ( r, θ ) = ( r · cos ( θ ) , r · sin ( θ ) , r 2 ) . Let D R 2 be the rectangle 2 r 0, 2 π θ 0. Let S = vector T ( D ).

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M 427L - Practice Final 1 - M 427L Practice for the third...

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