Vector Calculus Test I (Chapters 1-2) Review Questions:
Can I Answer These?
There are many concepts referred to below. If I have time to write up some sample problems
illustrating these, I will, but I cannot promise it. In the mean time, look back through
Vector Calculus Homework 1.5 (135 pts)
2.
In 8 show that:
t l# #lCt l# lB
t Ct l# lB
t Ct l#
(a) #lB
t Ct l# lB
t Ct l# B
t Ct B
t Ct B
t Ct B
t Ct
We have: lB
tB
t Ct B
t B
t Ct Ct Ct B
t B
t Ct B
t B
t Ct Ct Ct
B
#
#
t l #lCt l .
#lB
[10 pts]
t Ct l
Vector Calculus Homework 2.5 (300 pts)
2.
Prove that the following functions are differentiable, and find their derivatives at an
arbitrary point:
(a) 0 # p B C #
0 is differentiable because its partial derivatives, 0B ! 0C are
continuous everywhere, and
Vector Calculus Homework 1.3 (205 pts)
t , #3t t4 5t , and
4. Compute +t ,t -t where +t t3 #4t 5
-t $3t 4 #5 .
"
"
#
+t ,t -t ,t -t +t $ " # # $ " & ! (which
" # "
indicates that the volume of the parallelepiped spanned by these three
vectors is zero, so
example 12
1.2 The Inner Product, Length, and Distance 29
Physical forces have magnitude and direction and may thus be represented by vectors.
If several forces act at once on an object, the resultant force is represented by the sum
of the individual forc