Spring 2006 Final Review

# Spring 2006 Final Review - Math 53 - Final Exam Review GSI:...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 53 - Final Exam Review GSI: Santiago Canez http://math.berkeley.edu/∼scanez/courses/math53/spring06/ 1. Compute the line integral of F(x, y ) = (2x2 − 3y 2 )i + (2x + 3y 2 )j over the triangle with vertices (−2, 0), (2, 0), and (0, 2) oriented counterclockwise. 2. (a) Deﬁne the curl and divergence of a vector ﬁeld. (b) Determine whether or not the vector ﬁeld F = x2 y i + zyex j − 2xyz k is the curl of another vector ﬁeld. 3. Let S be the sphere of radius ρ. (a) Compute the surface area of S . ￿￿ (b) Compute the surface integral S z dS . 4. Let F(x, y, z ) = xi + y j + z k. Determine whether the ﬂux of F across S is negative, positive, or zero for each surface S below. (a) the cone z 2 = x2 + y 2 oriented outward (b) the sphere x2 + y 2 + z 2 = 1 oriented inward (c) the plane y = 2 oriented with normals having positive j component 5. Describe two diﬀerent methods for computing the ﬂux of F = 2yxi − yex j + (2 − z )k through the upwardly oriented piece of the plane x + y + 2z = 1 in the ﬁrst octant. 6. (a) Using Stokes’ Theorem, compute the surface integral of curl F over the upper unit hemisphere where F(x, y, z ) = xi − y j + exyz k with orientation given by normals with positive k component. (b) Using Stokes’ Theorem, compute the circulation of F(x, y, z ) = (z − 2y )i + (3x − 4y )j + (z + 3y )k around the circle x2 + y 2 = 4, z = 1 oriented counterclockwise. 7. Compute the ﬂux of F = x3 i + y 3 j + z 3 k through the sphere of radius 3, positively oriented. 8. Let S be the right half of the unit sphere, with inward orientation. Compute the ﬂux of F(x, y, z ) = zey i + (x2 + z 2 )j + sin−1 (exy )k across S . 9. Let S1 and S2 be two upwardly oriented surfaces with the same boundary and let F be a vector ﬁeld. Date : May 11, 2006 1 (a) Use Stokes’ theorem to show that the ﬂux of curl F through S1 is the same as the ﬂux through S2 . (b) Compute the ﬂux of curl F where F = −y i + xj + z k through the pyramid (without a base!) with vertices at (0, 0, 1), (±1, 0, 0), (0, ±1, 0), upwardly oriented. 10. Let C be the diamond with vertices at (±3, 0) and (0, ±3), oriented clockwise. Compute the line integral of y i − xj F= 2 x + y2 around C . Hint: Replace C (with justiﬁcation!) by a simpler curve. 2 ...
View Full Document

Ask a homework question - tutors are online