Coordinates Definition and 1000 Threads

1. Evaluate the double integral ∫∫arctan(y/x) dA by converting to polar coordinates over the Region R= { (x,y) | 1≤x^2+y^2≤4 , 0≤y≤x } My attempt at solving Converting to polar using x=rcosθ and y=rsinθ I get ∫∫arctan(tan(θ))r drdθ I understand that I have to integrate first with respect...

Integrate the function f(x,y,z)=−7x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt(263/137)x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20. I...

Homework Statement Find the volume using cylindrical coordinates bounded by: x2+y2+z2=2 and z = x2+y2 Homework Equations Converting to cylindrical coordinates: z = √2-r2 and z = r2 The Attempt at a Solution I figured z would go from r2 to √2-r2 r from 0 to √2 and θ...

Homework Statement Find the volume of the wedge-shaped region contained in the cylinder x^2+y^2=9 bounded by the plane z=x and below by the xy planeHomework Equations The Attempt at a Solution So it seems a common theme for me I have a hard time finding the limits of integration for the dθ term...

Homework Statement Evalutate the double integral sin(x^2+y^2)dA between the region 1≥x^2+y^2≥49 The Attempt at a Solution so r^2 = x^2 + y^2 dA = rdrdθ so I can turn this into double integral sin(r^2)rdrdθ where the inner integral integrated with respect to dr goes from 1 to 7...

Suppose you are observing the movement of an object on the Earth's surface. At any given moment, you know its current position (in lat/lon coordinates) and three prior positions. Each prior position is separated in time from the one after it by a small but variable number of seconds (say several...

Hi, I was wondering if anyone can help me. I don’t have a homework problem, but a problem I have encountered at work. I am a mechanical engineer working in the railway industry and I am struggling with a problem of reconstructing the vertical geometry of a rail in terms of height and...

Hello, I am currently reading about the topic alluded to in the topic of this thread. In Taylor's Classical Mechanics, the author appears to be making a requirement about any arbitrary coordinate system you employ in solving some particular problem. He says, "Instead of the Cartesian...

Homework Statement Suppose that in spherical coordinates the surface S is given by the equation rho * sin(phi)= 2 * cos(theta). Find an equation for the surface in cylindrical and rectangular coordinates. Describe the surface- what kind of surface is S? Homework Equations The...

Homework Statement Draw the graph of r = 1/2 + cos(theta) Homework Equations The equation is itself given in the question. It is a Limacon. The Attempt at a Solution Step-1 ---> Max. value of r is 1/2 + 1 = 3/2 [ at cos (0) ] Min. value of r is 1/2 - 1...

Homework Statement Say I am given a spherically symmetric potential function V(r), written in terms of r and a bunch of other constants, and say it is just a polynomial of some type with r as the variable, \frac{q}{4\pi\varepsilon_o}P(r), and we are inside the sphere of radius R, so r<R…...

I've been working on a problem where I have to find the geodesics for a given Riemannian Manifold. To present my doubt, I tried to find a simpler example that would demonstrate my uncertainty but the one I found, and shall present bellow, has actually a simplification that my problem doesn't, so...

Q: Consider the solid that lies above the cone z=√(3x^2+3y^2) and below the sphere X^2+y^2+Z^2=36. It looks somewhat like an ice cream cone. Use spherical coordinates to write inequalities that describe this solid. What I tried to do: I started by graphing this on a 3D graph at...

Hello MHB, So when I change to space polar I Dont understand how facit got $$\frac{\pi}{4} \leq \theta \leq \frac{\pi}{2}$$ Regards, $$|\pi\rangle$$ $$\int\int\int_D(x^2y^2z)dxdydz$$ where D is $$D={(x,y,z);0\leq z \leq \sqrt{x^2+y^2}, x^2+y^2+z^2 \leq 1}$$

I wondered anyone can explain the significance of the above as applied to metrics in the context of general relativity. This came up when the video lecturer in GR mentioned that r for example, was null or this or that vector or surface was null, say in the context of the eddington finkelstein...

Homework Statement Consider a planet orbiting the fixed sun. Take the plane of the planet's orbit to be the xy-plane, with the sun at the origin, and label the planet's position by polar coordinates (r, \theta). (a) Show that the planet's angular momentum has magnitude L = mr^2 \omega, where...

Dear All, To give a background about myself in Classical Mechanics, I know to solve problems using Newton's laws, momentum principle, etc. I din't have a exposure to Lagrangian and Hamiltonian until recently. So I tried to read about it and I found that I was pretty weak in coordinate...

show that \frac{d\hat{r}}{dt}=\hat{θ}\dot{θ} also, \frac{d\hat{θ}}{dt}=-\dot{θ}r i've tried finding the relationship between r and theta via turning it into Cartesian coord.s, and I've tried the S=theta r but still no luck. S=theta r dS/dt=d(theta)/dt r which is similar to the RHS...

Hi everyone, I would like to write the Laplacian operator in toroidal coordinate given by: $$ \begin{cases} x=(R+r\cos\phi)\cos\theta \\ y=(R+r\cos\phi)\sin\theta \\ z=r\sin\phi \end{cases} $$ where r and R are fixed. How do I do? More generally how do I find the Laplacian under a...

Homework Statement Two forces, vector F 1 = (4 i hat bold + 6 j hat bold) N and vector F 2 = (4 i hat bold + 8 j hat bold) N, act on a particle of mass 1.90 kg that is initially at rest at coordinates (+1.95 m, -3.95 m). A) What are the components of the particle's velocity at t = 10.3...

Homework Statement I need some help understanding a derivation in a textbook. It involves the Lagrangian in generalized coordinates. Homework Equations The text states that generalized coordinates {q_1, ..., q_3N} are related to original Cartesian coordinates q_\alpha = f_\alpha(\mathbf r_1...

Homework Statement Show that in cylindrical coordinates x = \rho cos \theta y = \rho sin \theta z = z the length element ds is given by ds^{2} = dx^{2} + dy^{2} + dz^{2} = d \rho^{2} + \rho^{2} d \theta ^{2} + dz^{2} Homework Equations -- The Attempt at a Solution...

I have a function to plot the orbits of planets based on their orbital elements (Semi-major Axis, Eccentricity, Argument of periapsis, Inclination, and longitude of ascending node). I have the x and y coordinates working great using only the semi-major axis, eccentricity, and argument of...

\nabla^2 u=\frac {\partial ^2 u}{\partial x^2}+\frac {\partial ^2 u}{\partial y^2}=\frac {\partial ^2 u}{\partial r^2}+\frac{1}{r}\frac {\partial u}{\partial r}+\frac{1}{r^2}\frac {\partial ^2 u}{\partial \theta^2} I want to verify ##u=u(r,\theta)##, not ##u(x,y)## Because for ##u(x,y)##, it...

Homework Statement Consider the function in polar coordinates ψ(r,θ,\phi) = R(r)sinθe^{i\phi} Show by direct calculation that ψ returns sharp values of the magnitude and z-component of the orbital angular momentum for any radial function R(r). What are these sharp values? The Attempt at a...

I am asked to compute the Curl of a vector field in cylindrical coordinates, I apologize for not being able to type the formula here I do not have that program. I do not see how the the 1/rho outside the determinant calculation is being carried in? Not for the specific problem - but for...

Homework Statement In some region of space, the electric field is \vec{E} =k r^2 \hat{r} , in spherical coordinates, where k is a constant. (a) Use Gauss' law (differential form) to find the charge density \rho (\vec{r}) . (b) Use Gauss' law (integral form) to find the total charge...

Homework Statement A particle initially located at the origin has an acceleration of vector a = 5.00j m/s2 and an initial velocity of vector v i = 8.00i m/s. a)Find the vector position at any time t (where t is measured in seconds). (Use the following as necessary: t.) Find the vector...

Homework Statement OK, we've been asked to derive the equations of motion in spherical coordinates. According to the assignment, we should end up with this: $$ \bf \vec{v} \rm = \frac{d \bf \vec{r} \rm}{dt} = \dot{r} \bf \hat{r} \rm + r \dot{\theta}\hat{\boldsymbol \theta} \rm + r...

Acceleration and coordinates at time t. Homework Statement At t = 0, a particle moving in the xy plane with constant acceleration has a velocity of vector v i = (3.00 i - 2.00 j) m/s and is at the origin. At t = 3.60 s, the particle's velocity is vector v = (7.00 i + 3.70 j) m/s. (Use the...

Problem: For the vector function \vec{F}(\vec{r})=\frac{r\hat{r}}{(r^2+{\epsilon}^2)^{3/2}} a. Calculate the divergence of ##\vec{F}(\vec{r})##, and sketch a plot of the divergence as a function ##r##, for ##\epsilon##<<1, ##\epsilon##≈1 , and ##\epsilon##>>1. b. Calculate the flux of...

Problem: Say we have a vector function ##\vec{F} (\vec{r})=\hat{\phi}##. a. Calculate ##\oint_C \vec{F} \cdot d\vec{\ell}##, where C is the circle of radius R in the xy plane centered at the origin b. Calculate ##\int_H \nabla \times \vec{F} \cdot d\vec{a}##, where H is the hemisphere...

Problem: Starting from the gradient of a scalar function T(x,y,z) in cartesian coordinates find the formula for the gradient of T(s,ϕ,z) in cylindrical coordinates. Solution (so far): I know that the gradient is given by \nabla T = \frac{\partial T}{\partial x}\hat{x}+\frac{\partial...

Problem: Rewrite the indefinite integral ## \iint\limits_R\, (x+y) dx \ dy ## in terms of elliptic coordinates ##(u,v)##, where ## x=acosh(u)cos(v) ## and ## y=asinh(u)sin(v) ##. Attempt at a Solution: So would it be something like, ## \iint\limits_R\, (x+y) dx \ dy =...

Homework Statement The formula for divergence in the spherical coordinate system can be defined as follows: \nabla\bullet\vec{f} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 f_r) + \frac{1}{r sinθ} \frac{\partial}{\partial θ} (f_θ sinθ) + \frac{1}{r sinθ}\frac{\partial f_\phi}{\partial...

Homework Statement Find the area inside the circle r = 3sinθ and outside the carotid r = 1 + sinθ The Attempt at a Solution Alright so I graphed it and found that they intersect at ∏/6 and 5∏/6. I can't think of a good way to approach the problem. The carotid has some of it's area...

Homework Statement Okay the graph SHOULD look like this. http://jwilson.coe.uga.edu/EMAT6680Fa11/Chun/11/21.png I can't make sense of this at all. It looks so weird. Why does it bend around the y-axis in such an asymmetric way? I just graphed r = sin(θ) with ease by making a table of r vs θ...

Homework Statement r=7sin(∅) find the center of the circle in Cartesian coordinates and the radius of the circle The Attempt at a Solution My math teacher is impossible to understand >.< and then the stupid homework is online and crap blah this class but I REALLY want to understand the material...

Homework Statement I have to turn this homework in online... I just want someone to check my work Convert from Cartesian coordinates to Polar coordinates (-1,-sqrt(3)) if r > 0 and if r < 0. Homework Equations The Attempt at a Solution if r > 0 then I believe the answer is...

Homework Statement I don't know how to make theta so ∅ = theta. find the slope of the tangent line at r = sin(6∅) when ∅ = pi/12 Homework Equations y=rsin(6∅) x=rcos(6∅) r=sin(6∅) tangent line equation y-y' = m(x-x') m = dy/dx The Attempt at a Solution when ∅ = pi/12 then...

Hi, can I say that a sphere is a plane, because in spherical coordinates, I can simply express it as <(r, \theta, \varphi)^T, (1, 0, 0)^T> = R? It does sound too easy to me. I'm asking because I'm thinking about whether it is valid to generalize results from the John-Radon transform (over...

Is partial derivative of ##u(x,y,z)## equals to \frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}+\frac{\partial u}{\partial z} Is partial derivative of ##u(r,\theta,\phi)## in Spherical Coordinates equals to \frac{\partial u}{\partial r}+\frac{\partial u}{\partial...

All but one of the tensor operations can be defined without reference to either coordinates or a basis. This can be done for instance by defining a ##(^m_n)## tensor over vector space ##V## as a multi-linear function from ##V^m(V^*)^n## to the background field ##F##. This allows us to define...

I've been studying Walter A. Strauss' Partial Differential Equations, 2nd edition in an attempt to prepare for my upcoming class on Partial Differential Equations but this problem has me stumped. I feel like it should be fairly simple, but I just can't get it. 10. Solve ##u_{x} + u_{y} + u =...

(a) $\vec{ST} = \pmatrix{9 \\ 9}$ so $V=(5,15)-(9,9)=(-4,6)$ (b) $UV = \pmatrix{-4,6}-\lambda \pmatrix{9,9}$ (c) eq of line $UV$ is $y=x+10$ so from position vector $\pmatrix{1 \\11}$ we have $11=1+10$ didn't know how to find the value of $\lambda$ (d) ?

Hi guys, This isn't really a homework problem but I just need a bit of help grasping rotations in spherical coordinates. My main question is, Is it possible to rotate a vector r about the y-axis by an angle β if r is expressed in spherical coordinates and you don't want to convert r...

Hi, Started to learn about Jacobians recently and found something I do not understand. Say there is a vector field F(r, phi, theta), and I want to find the flux across the surface of a sphere. eg: ∫∫F⋅dA Do I need to use the Jacobian if the function is already in spherical...

Hi ! I'm trying to inverse a mass matrix so I need to do something like this \dfrac{q}{\partial \mathbf{r}} where \cos q = \dfrac{\mathbf{r}\cdot \hat{\mathbf{k}}}{r} However, when \mathbf{r} = \hat{\mathbf{k}} \text{ or } -\hat{\mathbf{k}} I have problems. ¿What can I do...

wasn't sure about $\overrightarrow{AC}$

Homework Statement Hi everybody... i have a bad problem with my brain: starting from the Vectorial form of the magnetic dipole: \vec{B}(\vec{r}) =\frac{\mu_0}{4 \pi} \frac{3 \vec{r} ( \vec{r} \cdot \vec{m}) - r^2 \vec{m}}{r^5} Homework Equations i want to derive the spherical...