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Material Type: Exam; Professor: Li; Class: CALC SEVERAL VARIABLES; Subject: Mathematics ; University: Oklahoma State University - Stillwater; Term: Summer 2010;
Typology: Exams
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INSTRUCTOR: WEIPING LI
Print Name and Student #
(1) (10pts) (a) Sketch the vector field F(x, y) = (y, −x).
(b) Show that c(t) = (sin t, cos t, et) is a flow line of the velocity vector field F (x, y, z) = (y, −x, z).
(2) (10pts) Find the scalar curl of F(x, y) = yi − xj.
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2 WEIPING LI
(3) (10pts) Show that F = (x^2 + y^2 )i − 2 xyj is not a gradient field.
(4) (10pts) Show that V = (x + y^2 z^3 )i + ex^ cos(z^4 x^3 )j + (y^4 x − z)k can be the curl of some vector field F(x, y, z).
(5) (10pts) Change the order of integration, sketch the corresponding region, and evaluate thein- tegral. (^) ∫ (^4)
0
√x^ sin(y
(^3) )dydx.
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(7) (20pts) Let D be the region bounded by x = 0, y = 0, x + y = 1, x + y = 4. (a). Using the map T (u, v) = (u − uv, uv), find the region D∗.
(b) Find the determinant of the Jacobian for the map T.
(c) Evaluate
D x+^1 y dxdy.
VECTOR CALCULUS 5
(8) (10pts) Calculate
R(x^ +^ y)e
x−y (^) , where R is the region bounded by x + y = 1, x + y = 2 and x − y = 0, x − y = 1.
(9) (5pts) Evaluate
W
dxdydz (x^2 +y^2 +z^2 )^3 /^2 , where^ W^ is the solid bounded by^ x
(^2) + y (^2) + z (^2) = 1 , x^2 + y^2 + z^2 = 4 and z ≥ 0.
