{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

mm3 - 0 Q4[5 points Determine(giving reasons whether the...

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

Calculus IV [2443–004] Midterm III For full credit, give reasons for all your answers. Q1]. ..[15 points] Evaluate the following triple integral by ﬁrst sketching the region of integration, and then converting it to a spherical coordinates integral. Z 1 - 1 Z 0 - 1 - y 2 Z 2 - x 2 - y 2 x 2 + y 2 z dzdxdy Q2]. ..[20 points] Write down the equation in the statement of Green’s theorem, indicating what the various parts of it stand for. Compute R C F · d r directly, where F = h- x 2 y 2 ,xy i and C is the positively oriented boundary of the region bounded by the y -axis, the line y = 1, and the curve y = x . Use Green’s theorem to compute the path integral above by a second method. Compare your answers. Q3]. ..[20 points] State the fundamental theorem for path integrals. Let F = h ye yz cos( xy ) , ze yz sin( xy ) + xe yz cos( xy ) , ye yz sin( xy ) i . Show that curl ( F ) = 0 . Find a function f so that F = f . Use the fundamental theorem to give a quick computation of the path integral R C F · d r where C is the straight line curve from (0 ,e,π/ 2) to ( π, 1 / 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , 0). Q4]. ..[5 points] Determine (giving reasons) whether the following vector ﬁeld has positive, negative or zero divergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online