1. Solve the equation
and the initial condition
with
.
Sol.
2. Consider the wave equation subject to
and . Using the Fourier transform, show that the wave equation has a
solution of the form
cos .
3. Consider the two dimensional wave equ
1. Let be a point not on the plane that
passes through the points , and .
Show that the distance from the point to
where ,
the plane is
, and .
2. Use traces to sketch the quadric surface .
3. Let be a curve and be a point of curve.
(a) Find the uni
2010 (II) 2
1. Suppose that the temperature at a point in space is given by .
Find the rate of change of temperature at the point in the direction toward the point .
(a) In which direction does the temperature increase fastest at ?
(b) Find the maximum r
1. Evaluate the integral
,
where is the enclosed by the ellipsoid
.
2. Find the tangent plane to the surface with vector equation
sin cos , at the point .
3.(a) Evaluate the double integral
,
where , and is the disk .
(b) Evaluate the flux of the
06-07
1.
2.
3.
4. Let F cos i cos j sin k . Find a function such that F and evaluate the line
integral
F r, where
is the curve given by r isin j k .
5. Let .
(a) Show that the vector field
is conservative.
(b) Find a function such that .
(c) Evaluate
1
:
(2) 3
:
:
1. Let
.
(b) [4pts] Evaluate
the curve shown in the figure.
and for which the line
where is
(a) [6pts] Find the curve between the points
integral
,
reaches its minimum
value if it exists. If not, explain why.
: 120
2
: 100
2009 II 3
Let .
(a) Determine whether or not the vector field is conservative. If it is conservative, find a function
such that .
1.
(b) Find
where :
2. (a) Let be the region bounded by a positively oriented, piecewise-smooth, simple closed curve
2009 (II) 1
1.(a) Show that for any 3-dimensional non-zero vectors and , is orthogonal to .
(b) Show that , where and are non-coplanar vectors, and ,
. (Hint: , )
2. Find the volume of the tetrahedron with sides , , and .
3. Find parametri
2009 (II) 2
1. Let be a surface with equation and be a point on . Suppose is differentiable
and .
State and prove the equation of the tangent plane to the surface at a point .
2. Find parametric equations for the tangent line to the curve of intersection
06-07
1. Assume that and have continuous second-order partial derivatives. Show that
any function of the form is a solution of the wave equation
.
2. Suppose that over a certain region of space the electronical potential is given
by
.
(a) Find th
1. Let cos ln and
and .
(a) Find the derivative of at in the direction of
.
Sol.
(b) In what direction does increase most rapidly at ? What is the value of the
derivative in this direction?
Sol. ,
(c) Find an equation of the tangent plane to at
(2), (2) 1
1.[7pts] Let and be the lines with parametric equations
,
.
Find the distance between and .
2.[7pts] If and are (distinct) position vectors, show that the vector
is a normal vector to the plane through the points and .
3.[10pts] Show that