
Question (Feb 10):
I am confused at what we need to do for the first question #4, finding
the parametric representation for 4x^2 + 9y ^2 = 36. Also, I don't
know how to find z(t) for the integral questions.
Answer:
When you integrate, you have a function you are integrating (f(z),
e.g. f(z) = z bar in #18), and you also have a path that you are
integrating along.
The path can be given as an implicit equation (e.g. x^2+y^2=1), as a
parametric equation (e.g. x=cos t, y=sin t)), as a picture, or in
words ("the unit circle", "a square with corners 0, 1, i, 1+i").
No matter how to path is given, you first need to convert it to a
parametric form. To do this, you need to know how to convert the most
common paths to parametric form.
The most common paths are:
 unit circle
implicit: x^2 + y^2 = 1
parametric: x=cos t, y=sin t, where t=0..2pi
parametric complex: z=cos t + i sin t = e^{it}, where t=0..2pi
 circle centered at z0=x0+y0i with radius r
implicit: (xx0)^2 + (yy0)^2 = r^2
parametric: x = x0 + r cos t, y = y0 + r sin t, where t=0..2pi
parametric complex: z = x+iy = z0 + r e^{it}, where t=0..2pi
 elliple centered at the origin, with xintercepts +a,a,
yintercepts +b,b:
implicit: (x/a)^2 + (y/b)^2 = 1
parametric: x = a cos t, y = b sin t, where t=0..2pi
 curves of the form y=f(x)
implicit: y=f(x)
parametric: x=t, y=f(t)
 curves of the form x=f(y)
implicit: x=f(y)
parametric: x=f(t), y=t
 straight line from z0 to z1
parametric: z=z0 + t(z1z0), where t=0..1
 curves that consist of several pieces, e.g. 4 sides of a rectangle
Here you need to parametrize each of the pieces individually.
So for example, in question #4, you are dealing with the equation of
an ellipse, therefore 3) above applies. In #20, you are dealing with 4
sides of a square, so 7) applies: you parameterize each of the 4 sides
separately. Each is a straight line, so you use 6).
Path integrals and parametrization of paths are, by
the way, also covered in multivariable calculus, so these topics are a
repetition in this course. That's why we did not spend so much time on it in class.
