M 427L Practice for the third exam. The actual exam will not be so long
1. Let
vector
T
(
r, θ
) :
R
2
→
R
3
be a differentiable function. Which of the following is the
geometric meaning of
(
∂
vector
T
∂r
×
∂
vector
T
∂θ
)
·
dr
·
dθ
(a) This is a vector normal to the surface, with length (roughly) the area of
the image of a tiny rectangle of area
dr
·
dθ
.
(b) This is the axis of (infintesimal) rotation for flow of the vector field
vector
T
, its
length is (roughly) the angular velocity.
(c) This is a vector which points in the direction of maximal increase of
vector
T
(at
the point (
r, θ
)).
(d) This is the determinant of the Jacobian matrix for
vector
T
, and so gives the
factor by which
vector
T
stretches (infintesimally).
For 2 –3 let
vector
T
:
R
2
→
R
3
be the function
vector
T
(
r, θ
) = (
r
·
cos
(
θ
)
, r
·
sin
(
θ
)
, r
2
)
.
Let
D
⊂
R
2
be the rectangle 2
≥
r
≥
0, 2
π
≥
θ
≥
0. Let
S
=
vector
T
(
D
).
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 Spring '11
 staff
 Vector field, Stokes' theorem, dr · dθ

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