# Search results

1. ### The value of a Fourier series at a jump point (discontinuity)

Since ##f(0^+) = -\infty##, your function doesn't satisfy the Dirichlet conditions, so the usual theorem about convergence doesn't apply.
2. ### "Gabriel's Horn" - A 3-D cone formed by rotating a curve

Of course, if you use KNMP paint, there is no paradox. Definition: The xy plane is painted if for any ##r > 0##, the circular area at the origin of radius ##r## is covered. Theorem: A single drop of KNMP paint is sufficient to paint the ##xy## plane. Proof: Suppose a drop of KNMP paint has...
3. ### 3 Different and not parallel planes

For (b), think about a vector parallel to each of the parallel lines i.e., a common direction vector ##\vec D## for the intersection lines. What is the relation of the three normals to the planes to ##\vec D##? What does that tell you? For (c), What direction would ##\vec N_1 \times \vec N_2##...
4. ### Setting the limits of an integral

##r## going from ##0## to ##\cos\theta## is correct. I get ##\pi - 2## for the answer.
5. ### Problem in the parametrisation of a surface

##(\cos t, \sin t )## traverses the unit circle in the xy plane counterclockwise. If you want to go the other way, reverse the parameter, ##t \rightarrow -t##. What does that do to the expression?
6. ### Problem in the parametrisation of a surface

Imagine the area being actually cut and pulled slightly apart at the cut. Start to go around it. There's no way to walk around it without going the opposite direction on the inner circle. Just follow the arrows.
7. ### Problem in the parametrisation of a surface

It's a bit tricky to see geometrically. But imagine that cone was much shallower, almost flat. In fact, suppose in the extreme it was flat, and you are looking down on two concentric circles, with the normal pointing away from you (downward), By the right hand rule, the outer circle would be...
8. ### Applying Stokes' Theorem to the part of a Sphere Above a Plane

Let me add that when I read the OP's title and post, I assumed he wanted to verify Stoke's theorem, i.e., work both sides and show they are equal. If you can use Stoke's theorem, there is indeed an easy method to evaluate the flux integral, appropriate for a 3rd semester calculus class, which I...
9. ### Applying Stokes' Theorem to the part of a Sphere Above a Plane

I think you may be helping me make my point. No third semester calculus course I ever taught covered 3d rotations. I wouldn't expect a typical 3rd semester calculus student to know how to find the equations for your ##x',~y',~z'##.
10. ### Applying Stokes' Theorem to the part of a Sphere Above a Plane

Well, yes, but, playing Devil's advocate for the OP, assuming he knows how to make vectors ##\vec v_1,~\vec v_2##, the real problem is setting up the appropriate line and surface integrals with correct limits.
11. ### Applying Stokes' Theorem to the part of a Sphere Above a Plane

I think you have picked a random very tricky problem. First of all, the problem is stated poorly. I presume they are talking about the portion of the sphere above the plane. Note that the circular intersection of the plane and sphere is not a great circle and does not stay in the first octant...
12. ### Flux in a rotated cylindrical coordinate system

Did you check all the sides of the volume when you did the surface integrals? They each have their own normal and some of them may not be zero.
13. ### Flux in a rotated cylindrical coordinate system

I think your answer is correct. Also, looking at the answer choices, I would bet the first answer is supposed to be ##180##, given the pattern in the answer choices. Probably just a typo.
14. ### Solve this vector system containing sum and dot product equations

You didn't specify whether this is a 2D or 3D problem. You also didn't specify that ##\vec u, ~ \vec v,~ m## are given constants and the unknowns are ##\vec x## and ##\vec y##. Is that correct? Certainly, if it is a 3D problem there can be many solutions. For example if ##\vec v = \langle...
15. ### Using a Surface Integral for Mathematical Analysis of the Area of an Island

The shape of the island is an upside-down paraboloid. The integral looks like everything is OK except the ##r## limits should be reversed. You want to integrate from the smallest to largest values of ##r## to get a positive answer.
16. ### Find the vector equation of the line that passes through the point P and intersects with the straight lines R and S

I'm not quite sure what you actually want, but I don't think your equation for ##L_r## satisfies your definition for line ##r##.
17. ### Prove that this Function is a Homomorphism

What is the ##^*31## operation?
18. ### Changing Variables and the Limits of Integration using the Jacobian

Here's another way to see the new limits. Think about ##u## and ##v## in polar coordinates. Then ##u = r^2\cos^2\theta - r^2\sin^2\theta = r^2\cos(2\theta)## and ##v = 2r^2\cos\theta \sin\theta = r^2\sin(2\theta)##. Now think about a ray in the first quadrant at angle ##\theta##. Hold ##\theta##...
19. ### Changing Variables and the Limits of Integration using the Jacobian

I gather you have successfully changed the integral to ##u,~v## variables and are just asking about the limits. Am I correct about that? For doing the limits on this problem, what I have noticed is that ##u_x = v_y## and ##v_x = -u_y## suggesting that ##z = x^2 -y^2 + i(2xy)## is an analytic...
20. ### How to solve the integral which has limits from (1,2) to (2,4)

Your answer is not correct as you can verify that for your ##\phi##, ##\phi_x = P,~\phi_y = Q## fails. The problem is you have an extra ##\frac x y## in your answer. You really need to read the link I mentioned in post #7. In particular the last example.
21. ### Volume in the first octant bounded by the coordinate planes and x + 2y + z = 4.

The slanted plane leans up against the coordinate planes in the first octant. It is the only octant where the plane bounds a finite volume with the coordinate planes. So the 1/8 makes no sense. There is no larger volume that this is a portion of.

33. ### One-Dimensional Wave Equation & Steady-State Temperature Distribution

This reply is about using ##\{\sinh(kx), \cosh(kx)\}## pair instead of the ##\{e^{kx}, e^{-kx}\}## pair. When are solving ##y'' - k^2 y = 0##, with characteristic equation ##r^2 - k^2 = 0## and roots ##r = \pm k## that gives you an independent solution pair ##\{ e^{kx}, e^{-kx}\}## and you would...
34. ### Differentiate 𝑦 = (2𝑥^3 − 5𝑥 + 1)^20(3𝑥 − 5)^10

But it isn't in best factored form yet.
35. ### One-Dimensional Wave Equation & Steady-State Temperature Distribution

@Athenian : One thing that will simplify you work at this point is to use sinh and cosh functions instead of exponentials. So your ##Y_n(y)## would be ##C_n\cosh(k_ny) + D_n\sinh(k_ny)##. Then when you apply the boundary condition ##Y(0) = 0## you get ##C_n = 0##. So your solution will be a...
36. ### Surface integrals to calculate the area of this figure

I supose all those denominators that look sort of like an ##\alpha## are actually ##2##'s.Your figure shows the inner distance between the squares as ##1## unit and the outer distance between them is ##\frac 1 2 - (-\frac 1 2) = 1## so the squares have sides of length ##0##, so they aren't there.
37. ### Determine the distance between the following points and lines

Your graph is incorrect. The y intercept should be positive and the x intercept negative. And my answer (posting when tired) in post #1 (and yours) is incorrect . The correct answer should be ##\frac {21}{\sqrt{29}}##.
38. ### Determine the distance between the following points and lines

Have you drawn a graph? 13.6 looks quite a bit too big. I get about 6.8 with a different method. Edit: See below.
39. ### Wave Equation: d'Alembert solution -- semi-infinite string with a fixed end

It depends on the boundary conditions. For a string with fixed ends, you might want a periodic extension. The idea is if you have an infinite string where a point never moves, that is indistinguishable from the string being tied down at that point. Yes. It is that interference that gives what...
40. ### Wave Equation: d'Alembert solution -- semi-infinite string with a fixed end

You want to consider the odd extension of the function. Not the odd periodic extension. So your initial problem looks like an infinite string with initial displacement the odd reflection of the triangle and zero outside of ##[-2L,2L]## and released from rest. Solve that infinite string problem...

42. ### Verify Stokes' Theorem for this vector field on a surface

Yes. I inadvertently left out the ##r## going from ##0## to ##1##. And for the OP's benefit, I think he should get ##6\pi## for the answer to both.
43. ### Verify Stokes' Theorem for this vector field on a surface

What did you get for ##\nabla \times \vec A##? Think about parameterizing the surface with ##x = 3\cos\theta,~y=2\sin\theta## and similarly for the boundary curve.
44. ### Volume of an oblique circular cone

It's hard to tell from your graphic, but I'm not sure what you are describing is even a cone, slanted or not. Is the base curve a circle? Ellipse? Look at the following figure: You should have a base and the cross sections should be similar as in the figure. The volume of any such solid is the...
45. ### Factor ##a^6-b^6##

I would stop right there. It's factored.
46. ### Verify the convergence or divergence of a power series

No. The limit wasn't ##1##. It was ##|2x-1|\cdot 1##.
47. ### Verify the convergence or divergence of a power series

The ratio test gave you ##|2x - 1| < 1##. Factor out a ##2## and divide both sides by it giving ##|x - \frac 1 2| < \frac 1 2##. That tells you the center of the series is ##\frac 1 2## and the radius of convergence is ##\frac 1 2##.
48. ### Verify the convergence or divergence of a power series

Obviously a typo, he means 1/2.
49. ### Verify the convergence or divergence of a power series

It is pretty easy to show that both ##\frac{k+1}{k}## and ##\frac{\log(k+2)}{\log(k+1)}## go to ##1##.
50. ### 2-squareroot(16) = -2

All these answers to a 4 year old post by someone who hasn't been active for 2 years.