c Dr Oksana Shatalov, Spring 2013
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Spring 2013 Math 251
Week in Review 10
courtesy: Oksana Shatalov
(covering Sections 14.5-14.6 )
14.5: Curl and Divergence
Key Points
i
∂
• curlF = ∇ × F = ∂x
P
j
∂
∂y
Q
k ∂ ∂z R • If F is conservative, then curlF = 0.
• divF = ∇ · F
• div curl F = 0
1. Given the vector field F = zi + 2yzj + (x + y 2 )k.
(a) Find the divergence of the field.
(b) Find the curl of the field.
(c) Is the given field conservative? If it is, find a potential function.
R
(d) Compute C z dx+2yz dy +(x+y 2 ) dz where C is the positively oriented curve y 2 +z 2 =
4, x = 2013.
R
(e) Compute C z dx + 2yz dy + (x + y 2 ) dz where C consists of the three line segments:
from (0, 0, 0) to (2013, 0, 0), from (2013, 0, 0) to (2, 3, 1), and from (2, 3, 1) to (1, 1, 1).
2. Is there a vector field such that curlF = −2xi + 3yzj − xz 2 k?
14.6: Parametric surfaces and their areas
Key Points
• Parametric representation of surfaces: x = x(u, v), y = y(u, v), z = z(u, v),
(u, v) ∈ D.
• The graph of a vector-function of two variables, r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k,
surface.
(u, v) ∈ D, is a
• A normal vector to the tangent plane to a parametric surface at point (x0 , y0 , z0 ) = (x(u0 , v0 ), y(u0 , v0 ), z(u0 , v0 ))
is N(u0 , v0 ) = ru (u0 , v0 ) × rv (u0 , v0 ).
RR
RR
• Surface area: A(S) = S dS = D |ru × rv |dA.
3. Identify the surface x =
√
u cos v, y = 5u, z =
√
u sin v.
c Dr Oksana Shatalov, Spring 2013
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4. Identify the surface which is the graph of the vector-function r(u, v) = hu + v, u − v, ui.
5. Find a parametric representation of the following surfaces:
(a) x + 2y + 3z = 0;
(b) the portion of the plane x + 2y + 3z = 0 in the first octant;
(c) the portion of the plane x + 2y + 3z = 0 inside the cylinder x2 + y 2 = 9;
(d) z + zx2 − y = 0;
(e) y = x2 ;
(f) the portion of the cylinder x2 + z 2 = 25 that extends between the planes y = −1 and
y = 15
6. Find an equation of the plane tangent to the surface x = u, y = 2v, z = u2 + v 2 at the point
(1, 4, 5).
7. Given a sphere of radius 2 centered at the origin, find an equation for the plane tangent to
√
it at the point (1, 1, 2) considering the sphere as
p
(a) the graph of g(x, y) = 4 − x2 − y 2 (Note that the graph of g(x, y) is the upper halfsphere );
(b) a level surface of f (x, y, z) = x2 + y 2 + z 2 ;
(c) a surface parametrized by the spherical coordinates.
8. Find the area of the part of the cylinder x2 + z 2 = 1 which lies between the planes y = 0
and x + y + z = 4.
9. Find the area of the portion of the cone x2 = y 2 + z 2 between the planes x = 0 and x = 2.