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Unformatted text preview: ——l Cur1(cur1F)
) yes Name & ID: W 400500 04/24/2009 Responses without work will receive no credit. The exercises on the ﬁrst page are Worth 1
point each and the problem on the second page is worth two points. Exercise 1. Let f be a scalar ﬁeld (a function from R3 to R) and F a vector ﬁeld (a function
from R3 to R3), and assume both are smooth (so that all partial derivatives of all orders exist).
Though it is possible to give meaning to gradF (the result would be .a function which produces
3 X 3 matrices), for the sake of this problem (and this course) we will (not allow grad F. Fill in the
following table. ' Expression Makes sense? "Result (scalar
(yes/no) or vector) curlcgradf —l div<cur1eradm —mz Exercise 2. Suppose I tell you F is a conservative vector ﬁeld on a connected open region I), D contains the point ((1,3)), and I deﬁne a function f (x, :0) on D by f($,y)=/OFdr where C' is any smooth oriented path that begins at ((1,3)) and ends at (say). Does this deﬁne a
function f (:13, 3;)? (That is, does the deﬁnition make sense and does the result pass the vertical; line test?) What if I remove the assumption that D is connected? rm Dewmwoﬂw MATH 53 Diacusaion_— Kaspar Exercise 3. Evaluate the line integral two different ways, once as a line integral directly and once
using Green 5 Theorem: fig? dm—ixydy,
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 Summer '09
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