1. Let cos ln and
(a) Find the derivative of at in the direction of
(b) In what direction does increase most rapidly at ? What is the value of the
derivative in this direction?
(c) Find an equation of the tangent plane to at
1. Solve the equation
and the initial condition
2. Consider the wave equation subject to
and . Using the Fourier transform, show that the wave equation has a
solution of the form
3. Consider the two dimensional wave equ
sample space & sigma field
Cond. Expect. of X given F :
The best reasonable
prediction of X
F = (Y) : E[X|Y]
E[X|Y](w) = E[X|Y (Y(w)]
E[X|B] = 1/P(B) BXdP
- Basic Stochastic Processes, Brzezniak and Zastawniak
properties of distributions
of future increments
w.r.t. the past
E[ future | p
(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
1. Let be a point not on the plane that
passes through the points , and .
Show that the distance from the point to
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
4. Let F cos i cos j sin k . Find a function such that F and evaluate the line
F r, where
is the curve given by r isin j k .
5. Let .
(a) Show that the vector field
(b) Find a function such that .
(b) [4pts] Evaluate
the curve shown in the figure.
and for which the line
(a) [6pts] Find the curve between the points
reaches its minimum
value if it exists. If not, explain why.
2009 II 3
(a) Determine whether or not the vector field is conservative. If it is conservative, find a function
such that .
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
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
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
(a) Find th