# To the rest of r a what is the general form of the

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to the rest of R . (a) What is the general form of the Fourier series for f ? (b) Give the formulas for the Fourier coefficients of f . 4. [6 points] Give a parametrization of the curve of intersec- tion of the plane y + z = 1 and the sphere x 2 + y 2 + z 2 = 3, oriented in the counterclockwise direction when viewed from above.

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MATB42H page 2 5. [15 points] Let γ be the curve parametrized by γ ( t ) = ( t, ln t, 2 2 t ) , 1 t 2. (a) Find the length of the curve γ . (b) Evaluate integraldisplay γ f ds , where f ( x, y, z ) = x + z 2 . (c) Evaluate integraldisplay γ F · d s , where F ( x, y, z ) = ( x, x 2 y, xz ). 6. [15 points] Let γ ( t ) be a parametrization of the boundary curve of the region bounded by y = x 2 - 1 and y = x + 1, oriented in the clockwise direction. Evalu- ate each of the following line integrals over γ . (a) integraldisplay γ ( e y + 2 xy cos( x 2 y ) ) dx + ( xe y + x 2 cos( x 2 y ) ) dy (b) integraldisplay γ F · d s , where F ( x, y ) = parenleftbigg - y x 2 + y 2 , x x 2 + y 2 parenrightbigg (c) integraldisplay γ ( xy + x cos x 2 ) dx + ( 2 x + y sin y 2 ) dy 7. [5 points] Determine the flow lines of the vector field F ( x, y ) = (1 , x 2 ).
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