47 MIDTERM 2B

# 47 MIDTERM 2B - ρ T θ =-ρ sin φ sin θ i ρ sin φ cos...

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1 MIDTERM II NAME MATH 20E (please print) TA’S NAME (please print) FORM B Problem 1. (14 pts.) For the vector ﬁeld F = xi + zyj + zk ﬁnd: (a) div F (b) curl F Problem 2. (14 pts.) Compute the iterated integral: Ú 1 0 Ú y 0 (20 y 2 x + cos π 2 x ) dxdy

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2 Problem 3. (14 pts.) By use of triple integrals compute the volume of the solid: x 2 + y 2 + z 2 ± 4 and z ² 0 . Problem 4. (14 pts.) Compute the following line integral over the curve C given by parametric equation c ( t ) = (cos t, cos t, sin t ) , 0 ± t ± π 2 : Ú C ydx - xdy + zdz
3 Problem 5. (14 pts. ) Find the equation of the tangent plane to the sphere x 2 + y 2 + z 2 = 1 at the point (0 , - 1 , 0) . Problem 6. (15 pts) Find the surface area of the cylinder: x 2 + y 2 = 4 , 0 ± z ± 1 .

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4 Problem 7. (15 pts) Let S be the surface of the cone: x 2 + y 2 = z 2 , 0 ± z ± 1 . Find the following integral over the surface S : Ú Ú S z 2 dS
5 FORMULA SHEET 1. Spherical coordinates: x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, dxdy dz = ρ 2 sin φdρdθ dφ Parametrization of the sphere with radius
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Unformatted text preview: ρ : T θ =-ρ sin φ sin θ i + ρ sin φ cos θ j + 0 k , T φ = ρ cos φ cos θ i + ρ cos φ sin θ j +-ρ sin φ k T θ × T φ =-ρ 2 sin 2 φ cos θ i-ρ 2 sin 2 φ sin θ j-ρ 2 sin φ cos φ k . || T θ × T φ || = ρ 2 sin φ. 2. Cylindrical coordinates: x = r cos θ, y = r sin θ, z = z, dxdy dz = r dr dθ dz Parametrization of the cylinder with radius r T θ =-r sin θ i + r cos θ j + 0 k , T z = 0 i + 0 j + k T θ × T z = r cos θ i + r sin θ j + 0 k . || T θ × T z || = r. 3. Parametrization of the cone x 2 + y 2 = z 2 : x = z cos θ y = z sin θ z = z. T θ =-z sin θ i + z cos θ j + 0 k , T z = cos θ i + sin θ j + k T θ × T z = z cos θ i + z sin θ j-z k . || T θ × T z || = √ 2 z....
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47 MIDTERM 2B - ρ T θ =-ρ sin φ sin θ i ρ sin φ cos...

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