Spring 2003

# Mathematics Math21a Spring 2003

## Multivariable Calculus

Course Head: Oliver knill
Office: SciCtr 434
Email: knill@math.harvard.edu
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## Frequently asked questions

 Send questions of potential general interest to math21a@fas.harvard.edu.
 Question: When is the project due? Answer: At the last day of class.
 Question: I have a question to the grading of the exam. Answer: See or inform your section leader immediatly. More than 3 days after the receiption of the exam, no regrade requests are possible anymore. (You are however welcome to ask questions also later about exam questions.)
 Question: How many challenge problems have to be completed for the project? Answer: 20 stars * correct give full credit.
 Question: Can I do both the challenge problems as well as the Mathematica project. Answer: Yes, but you will have to decide which of the two you want to turn in.
 Question: Why can one sometimes describe a surface with a function of two variables, and sometimes as a function of three variables. Answer: Many surfaces are described as { (x,y,z), g(x,y,z) = 0 } like for example the sphere x2+y2+z2-1 = 0. A special case is when g(x,y,z) = z-f(x,y). In that case, g(x,y,z) = 0 means z = f(x,y) and the surface is the graph of a function of two variables.
 Question: Can we submit more than 20 stars' worth of challenge problems (i.e., have backup problems in case one or two are incorrect)? Also, is partial credit given on some problems? Answer: You can submit as many challenge problems as you wish. If you have 20 stars correct, this gives full credit, independent of whether there are some mistakes in other submitted answers. We also acknowledge partial credit.
 Question: When finding a volume of something, when is it a good time to take a double integral and when is it good to take a triple integral? Answer: the volume is always a triple integral. The volume is the triple integral of f(x,y,z)=1 over a region in space. In the special case, when the region is bounded above by the graph of a function g(x,y) and the xy plane, then the triple integral is ``` int int_R [ int_0^g(x,y) 1 dz ] dx dy = int int_R g(x,y) dx dy ``` The last double integral is actually a triple integral for which one integration has already been done. The situation is the same one dimension lower. The area of a region R is always the double integral ``` int int_R 1 dx dy ``` But in the special type I case, where the region is bounded above by the graph of a function g(x), then the double integral is ``` int_a^b int_0^g(x) dy dx = int_a^b g(x) dx ``` This single ingegral is actually a double integral for which the integration over the y variable has been done already.
 Question: What is the difference in finding the area and the surface area? Answer: To find the area of R, you integrate the constant function f(x,y) = 1 over the region R. This is a double integral. To find the surface area of a surface S parameterized by r(u,v)=(x,y,z), you integrate the function f(u,v) = |r_u x r_v| over R. Also this is a double integral. You can distinguish these two cases. In the first case, you work with a region in the plane. In the second case, you deal with a surface S in space (which happens to be parametrized by a region in the plane and allows you to use a double integral).
 Question: I'm getting confused with directional derivatives and gradients. Gradients are perpendicular to the curve if parametrized in 2 variables or perpendicular to the surface when given in 3 variables. So why is vector ru(u,v) tangent to the surface? Why isn't it perpendicular? Answer: ru is a vector tangent to the surface because it is the velocity vector of the curve R(u) = r(u,v), called a grid curve. Also rv is tangent. The normal vector would be the cross product of n = ru x rv. Describing a surface as a parametrized surface with a function r(u,v) is one way to define a surface. An other way is to give it as a level surface g(x,y,z)=c of a function of three variables g. In this case, called implicit description, the normal vector is the gradient vector n = (gx,gy,gz). These two ways to describe a surface are quite different. In the case of a sphere for example, the first way would be r(u,v) = (L cos(u) sin(v), L sin(u) sin(v), L cos(v)) the second would be g(x,y,z)=L2 with g(x,y,z) = x2+y2+z2. Switching between the parametrized description and the implicit description is often not so easy. It can be done in some cases like the plane, sphere, ellipsoid, cone, or graphs of functions f(x,y) of two variables. In the latest case, we can either say g(x,y,z) = 0 with g(x,y,z) = z-f(x,y) or give the parametrizaion r(u,v) = (u,v,f(u,v)).