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so33lutions

so33lutions - -——l Cur1(cur1F yes Name& ID W 400-500...

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Unformatted text preview: -——l Cur1(cur1F) ) yes Name & ID: W 400-500 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)=/OF-dr 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—i-xydy, c Where C IS the triangle with corners (0,) O), (1 ,,(1) We WM» W( M W” (650 05 (1,1)- I C? C rue-5&5) 05,55 1 C,- (00) , C7. « flu-:3“ l't/h 05f-‘ C? LT; (1:):(( 49 Fri?!) 64 f4, ' M f IPA47+CQ0LU\$ J: P(Xm,lﬂ))x )‘(t Jrﬂxlllqﬁyﬁ} Lgm’xw Mﬁwmég 3:52; fﬁﬂv‘rwwﬁuﬂ (—5005 1 Mung Lgfﬂﬂdﬂ focamm 5d) 0W5. TIM/3%: ﬂM+Xj5%% : “J5 31,352, 555.155P 55, ....5 .__. ‘ I 1 ::l I 7 5:- mm, 35M 555, 5,555,055, :55 _ _ , 1 ' ' . {\$0 '71 7 27K ,7 5.7‘ X 6.x x20 77 779117727 '7 77137 ...
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so33lutions - -——l Cur1(cur1F yes Name& ID W 400-500...

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