SM311o { First Exam
Sept 27, 1995
1. Let be a point in 3 with cartesian coordinates (1 2 3). Find the
spherical coordinates of this point.
2. Let ( ) = 2 3 + 3 2 .
(a) Find the gradient of at = (1 2).
(b) Find the directional derivative of at = (1 2) in the direction
e = h1 ,1i.
(c) Find the direction of maximum ascent at = (,1 3).
(d) Write down the Mathemtica command that plots over a typical
domain.
3. Let ( ) = 2 2 + 2 .
(a) Draw two of the typical level curves of this surface.
(b) Let be the 1-level curve of this surface. Show that the point =
( 21 p12 ) is on this level curve. Determine the gradient of at . Draw
the graph of together with the point and its gradient. What is
the geometric relationship between the gradient vector and ?
(c) Write the Mathematica command that draws the level curves of .
4. Let v = h px2y+y2 , px2x+y2 i be the velocity eld of a uid.
P
R
f x; y
x y
;
;
x y
f
P
;
f
P
;
;
P
;
f
f x; y
x
y
C
P
;
f
C
P
P
C
f
;
(a) Show that v is incompressible.
(b) Let ( ( ) ( )) be the particle path of a typical particle in the uid.
Write down the system of dierential equations one must solve to
determine ( ) and ( ).
5. Let v = h , + i. Does v have a stream function? If no, explain why.
If yes, determine it.
6. Let and be two smooth functions of the variables and . Prove the
identity
r( ) = 12 ( r , r )
x t ;y t
x t
y
f
x; x
y t
y
g
x
f
g
g
1
g
f
f
g
:
y