Frequently asked questions
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Question:
Are the challenge problems extra credit or are they
simply intended to supplement and reinforce the
chapter questions?
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Answer:
Doing the challenge problems is one the section project options:
- Do a Mathematica (Computer algebra) project.
- Do a project from the book.
- Do some challenge problems.
- Do an in class (or takehome) test around the 12'th week.
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Question:
Why can a variable be anything, when it is not specified?
Example: in the equation x2+y2=1, the variable z does not appear.
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Answer:
In your example, only the x and y variable are constrained. The z value can
be arbitrary. The points satisfying the equation x2+y2=1
form a R2, but since any z value will still allow x and y to satisfy the
conditions, in R3 you actually get a cylinder.
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Question:
What really is the difference between <1,1> and (1,1)? I know one is a vector and one
a point, but don't they signify the same thing?
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Answer:
The notation difference is just to distinguish vectors and points.
A vector attached at origin defines a point. So the vector <1,1,1> extends from the origin (0,0,0)
to the point (1,1,1).
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Question:
I can't keep all the vectors and scalars straight--if you multiply two of them
together, what do you get?
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Answer:
There are a number of things that can occur when multiplying vectors and scalars:
- A scalar times a scalar is still a scalar
- A scalar times a vector is a vector
- A vector times a vector can be either a scalar (when doing the
dot product) or a vector (when doing the cross product).
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Question:
In number 34, section 9.2, does the |r-r1| denote
the absolute value or magnitude of the vector?
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Answer:
They represent the magnitude (length) of the vector going from r to r1.
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Question:
Why is the gradient of f(x,y) not perpendicular to the surface defined by f(x,y)?
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Answer:
The gradient of f is perpendicular to the level sets of f, not the graph of f. To get
a vector perpendicular to the graph, you write the graph first as a level set g(x,y,z)=z-f(x,y)=0.
Then, grad(g) is perpendicular to the graph.
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Question:
When do I use Lagrange? When do I plug in the boundary? When do I just take the gradient and set it
equal to zero?
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Answer:
Finding extrema of a function appears in two cases.
If there are no restrictions on the function, find the critical
points and use the second derivative test.
If the extrema have to be found under the condition that some
other equation g(x,y)=0 is satisfied, then use Lagrange.
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Question:
What does stretching factor mean and when do I need them?
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Answer:
Th "stretching factor" is a name used to denote something you add when switching variables
in integration.
All you need to know for 21a is that if you
switch to polar coordinates you need to change dxdy to r drd(theta) (the streching factor is r),
in spherical coordinates, the stretching factor is
(rho)2sin(phi)d(rho)d(phi)d(theta). When integrating over surfaces, the streching
factor is |ru x rv|.
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Question:
What is a vector field?
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Answer:
A vector field defines a vector for each point in the plane or space like
F(x,y,z) = (P(x,y,z),Q(x,y,z),R(x,y,z)). Vector fields appear in physics for
example as velocity fields of fluids, as force fields, as electric or magnetic
fields.
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Question:
How do I see, when to switch from rectangular to polar, cylindrical, or spherical coordinates?
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Answer:
You often see it from the form of the region and if there are terms in the function which
simplify in s specific coordinate system. If you can not solve a multiple integral in
specific setup, it is a good idea to try to solve it in an other coordinate system (besides trying
to switch the order of integration).
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Question:
When do I use triple integrals to get the volume and when do I use double integrals.
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Answer:
A volume is always a triple integral. If you compute the volume under the graph of a
function z=f(x,y), then only a double integral is needed (this double integral is actually
a triple integral in disguised form).
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Question:
Is there an easy order of integration for non-rectangular coordinates? And, when can you simply
switch the order?
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Answer:
As in rectangular coordinates, there is no prefered order.
If you can not solve an integral in one order, try to switch
the order of integration.
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