Finding Solutions for Intricate Homework Problems

In summary, the author solves the line integral equation for r around a path in terms of the curve parameter. He uses the relation dr = ddr/d\theta to rewrite the differential.
  • #1
leonardthecow
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Homework Statement


Hi there! So I'm working on an old homework problem for review so that I actually have the solution, but I'm not sure how to compute a certain part. Here it is:

$$\int \int rcos^2(\theta)dr - r^2cos(\theta)\sin(\theta)d\theta$$

Homework Equations


The solution involves (what I think is) a rewrite of $$dr$$ in terms of $$d\theta$$, but I don't follow the substitutions the author makes.

The Attempt at a Solution


Any thoughts on how to approach a problem like this? Thanks!
 
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  • #2
leonardthecow said:

Homework Statement


Hi there! So I'm working on an old homework problem for review so that I actually have the solution, but I'm not sure how to compute a certain part. Here it is:

$$\int \int rcos^2(\theta)dr - r^2cos(\theta)\sin(\theta)d\theta$$

Homework Equations


The solution involves (what I think is) a rewrite of $$dr$$ in terms of $$d\theta$$, but I don't follow the substitutions the author makes.

The Attempt at a Solution


Any thoughts on how to approach a problem like this? Thanks!

You have titled this thread "line integral". But it doesn't look like a line integral since it is a double integral. You haven't given us a statement of the problem or any equations to work with. We aren't mind readers.
 
  • #3
Well, the original problem is to compute the line integral of $$v= r\cos^2(\theta)\hat{r} -r\cos(\theta)\sin(\theta)\hat{\theta} +3r \hat{\phi}$$ around a path depicted in the text. The path, in terms of spherical coordinates, runs as follows, broken into 3 segments:

1) $$r:0 \rightarrow 1, \theta = \frac{\pi}{2}, \phi = 0$$

2) $$r=1, \theta= \frac{\pi}{2}, \phi: 0 \rightarrow \frac{\pi}{2}$$

3) $$r: 1 \rightarrow \sqrt{5}, \phi = \frac{\pi}{2}, \theta: \frac{\pi}{2} \rightarrow \arctan(1/2)$$

The integral comes from computing the dot product of v with dl along the third path. I didn't inclue the rest of the problem since my question is one of how to reduce th double integral to a single integral in theta, as the author did in the solution.
 
  • #4
Assuming that (3) is a straight line, you can parametrize it using the angle ##\theta## as the curve parameter. You can then use the relation ##dr = \frac{dr}{d\theta} d\theta## (essentially the chain rule) to rewrite the differential. Note that what you want for ##r(\theta)## is a linear function such that ##r(\pi/2) = 1## and ##r(\arctan(1/2)) = \sqrt 5##.
 
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  • #5
Ah thank you! That's the step I was missing, 3 is a straight line. This is exactly what I needed.
 

1. What is the best approach for tackling intricate homework problems?

The best approach for solving intricate homework problems is to break them down into smaller, more manageable tasks. Start by identifying the main problem and then breaking it down into smaller sub-problems. This will help you focus on one aspect at a time and make the problem more approachable.

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To improve your problem-solving skills for intricate homework problems, practice is key. The more you practice, the more you will become familiar with different problem-solving techniques and strategies. Additionally, seeking help from teachers, classmates, or online resources can also help you learn new approaches to solving problems.

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4. What should I do if I get stuck on an intricate homework problem?

If you get stuck on an intricate homework problem, take a break and come back to it later with a fresh perspective. Sometimes, stepping away from the problem for a bit can help you see it in a new light. You can also try approaching the problem from a different angle or seeking help from a teacher or classmate.

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