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Unformatted text preview: Math 2220 Section 3.3 : Problem Set 9 Spring 2010 1. For the vector ﬁeld F(x, y ) = (−x, y ) do the following things: (a) Sketch the vector ﬁeld. (b) Determine the ﬂow lines by solving the diﬀerential equations dx = −x and dy = y . dt dt Your answer should be functions x(t) and y (t) each of which depends on an arbitrary constant. (c) Check that the ﬂow lines are hyperbolas by eliminating the variable t from your formulas in part (b). (d) Find the values of the arbitrary constants that give the ﬂow line c(t) such that c(0) = (2, 1) . 2. Sketch the vector ﬁeld F(x, y ) = (x, x2 ) , determine the ﬂow lines, and show the ﬂow lines are parabolas. 3. Verify that the parametrized curve c(t) = (sin t, cos t, 2t) is a ﬂow line for the vector ﬁeld F(x, y, z ) = (y, −x, 2) . 4. Consider the vector ﬁeld F(x, y ) = (2x, −3) . (a) Find a function f (x, y ) whose gradient vector ﬁeld is equal to F , so F = ∇f . (b) Determine the equipotential curves of F (in other words, the level curves of f ). Sketch a couple of these curves, and sketch what the vector ﬁeld F looks like along these curves. Section 3.4 : 1. Compute the divergence and the curl for the following vector ﬁelds: (a) F(x, y ) = (x2 + y 2 , x2 − y 2 ) √ 2 2. Compute the divergence of the vector ﬁeld F(x, y, z ) = (z cos(ey ), x z 2 + 1, e2y sin 3x) . 3. Show that the vector ﬁeld F(x, y ) = (x2 + y 2 , −2xy ) is not the gradient ﬁeld of any function by showing that curl(F) is nonzero. 4. Show that each of the following vector ﬁelds F have curl(F) = 0 by using the general fact that curl(∇f ) = 0 and ﬁnding a speciﬁc function f such that F = ∇f . (a) F(x, y, z ) = (yz, xz, xy ) . (b) F(x, y ) = (3x2 y, x3 + y 3 ) . 5. Verify the identity ∇ × (f F) = f ∇ × F + ∇f × F for functions f and vector ﬁelds F . (b) F(x, y, z ) = (x + y, y + z, x + z ) . (c) F(x, y, z ) = (x, y 2 , z 3 ) . ...
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This note was uploaded on 04/21/2010 for the course MATH 2220 taught by Professor Parkinson during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 PARKINSON
 Equations, Multivariable Calculus

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