Polar Definition and 1000 Threads

Just starting up school again and having trouble remembering some mathematics. Here's the problem. Find the inner product of ⃗a = (1, 45◦) and ⃗b = (2, 90◦), where these vectors are in polar coordinates (r, θ). Thanks =) 1st post here btw.

Hi, I'm trying to find the area of a segment of a circle that is not at the origin. It will look similar to this picture below but I need to find the area enclosed by a circle. Using the polar equation of a circle provided by wikipedia: and integrating to find the area of a...

Hello all. I am trying to change: E = (1/r) ar To rectangular coordinate system. Where ar is a unit vector. So I know r = √(x^2 + y^2) i also think ar = ax+ay, where ax and ay are unit vectors along the x-axis and y-axis respectively. So that would give me: E = (1/√(x^2 + y^2)) (ax...

Hello. I have 2D Euler equation for fluids. I can't derive it in polar coordinates. I defined functions u(x,y,t) = u'(r, theta, t) and v(x,y,t) = v'(r, theta, t). I started by computing derivatives \frac{\partial u'}{\partial r}=\cos\theta\frac{\partial u}{\partial...

Not homework as such, just need some clarification. When finding \alpha do you have to take the signs into account when finding tan^{-1} x/a. Does it matter if a or x are negative? Next question is about quadrants 1: \theta = \alpha 2: \theta = \pi - \alpha 3: \theta = -\pi -...

If we divide the polar curve into infinitely thin sectors, the arc length of a single sector can be approximated by ds = \frac{dθ}{2π}2πr = rdθ. So why can't we model the arc length of the curve as \int^{β}_{α} rdθ It turns out that the correct formula is actually...

Homework Statement from the cartesian definition of angular momentum, derive the operator for the z component in polar coordinates L_z = -ih[x(d/dy) - y(d/dx)] to L_z = -ih(d/dθ) Homework Equations x = rcosθ y = rsinθ r^2 = x^2 + y^2 r = (x^2 + y^2)^1/2 The Attempt at...

Homework Statement r=5^theta theta goes from 0 to 2Pi Homework Equations Length= integral between a and b of sqrt(r^2+(dr/dtheta)^2)dtheta The Attempt at a Solution r^2=25^theta or 5^(2theta) dr/dtheta=5^theta (ln 5) (dr/dtheta)^2=25^theta+10^theta (ln 5)+...

Homework Statement Express the following equation in Cartesian form r = 1 - cos(θ) Homework Equations x = r*cos(θ) y = r*sin(θ) r^{2} = x^{2} + y^{2} tan(θ) = \frac{y}{x} The Attempt at a Solution I have no idea... a hint would be nice thanks! BiP

Homework Statement The problem is attached in the picture. The Attempt at a Solution This looks like a vector in polar notation to me (r, θ). But the application of vectors to phases and periodicity is not even mentioned in the chapter! (Vector Algebra) I've tried to make sense of...

I was given the problem r=2sin(2(θ)). I'm supposed to write the equation in the Cartesian Coordinates. I understand the basics to this but I'm not really sure how I'm supposed to write the equation when I have x=2sin(2(θ))cos(θ) and y=2sin(2θ)sin(θ).

Are ω and \dot{θ} the same in a polar kinematics? I know ω is angular speed (rad/s) and it seems to me that \dot{θ} would be the same, but in the context of rotation in polar coordinates where v = \dot{r}\widehat{r}+ r\dot{θ}\widehat{θ}, v = rω, and vθ = r\dot{θ}, that doesn't seem to be...

Homework Statement Find the area of the region that lies inside both of the circles r = 2sin(x) r = sin(x) + cos(x) Homework Equations A = (1/2)(int from a to b): r^2 dx (I apologize because I do not know how to make calculus look proper in text form) The Attempt at a Solution...

Hi, I have a pretty simple question but I'm not certain I know how to phrase it properly. I will try. When we are integrating using cartesian coordinates to find the area under a curve, area under the x-axis is negative and area above the x-axis is positive. This makes sense when I...

The question I have is a bit strange. I do not have ANY formulas or equations given. I was only given bunch of points with r, theta, and z. Z being the depth. R being radius and Theta being the angle. I was wondering if there is a way to find a rough estimate volume of the following. R...

The solution to a linear differential equation is, y=exp(ax). If a is complex ,say a=b+ic, then the period is T=2pi/c. My question is, if a is in polar form, a=r*exp(iθ), how is the period then T=2pi/θ. Any help would be great, Thank, Will

When plotting graphs in polar coordinates, how does one know when to make the graph sharp (at θ=0) (as in for the graph for r=1-cosθ) as opposed to a dimple (r=3/2 + cos θ) ?

Homework Statement Integrate y/(x^2+y^2) for x^2+y^2<1 and y> 1/2 ; use change of variables to polar coordinates Homework Equations THe above The Attempt at a Solution the variables transform as y=rsinz x=rcosz, where z is an angle between pi/6 and 5*pi/6 = which is the...

Hello, I recently run into a problem. Let's say I have the point (a,b) and (-a,-b). The, I know that θ_1 = atan(b/a) and θ_2 = atan((-b)/(-a)) = θ_1. But, what if I want to go back to Cartesian coordinates? If I assume r = 1, a = cos(θ_1) and b = sin(θ_1) while -a = cos(θ_2) and b =...

Homework Statement I'm not grasping how to convert a surface with known rectangular graph to a parametric surface (using some polar techniques, I assume). I would appreciate it if someone could clarify the conversion process. One of the examples is as follows: A sphere...

Homework Statement I was wondering if I did this problem correctly as I don't have the solution, also wanted to make sure that my limits of integration were correct as they tend to be tricky in finding arc length in polar coordinates. x(t)=arcsint y(t)=ln(sqrt(1-t^2)) Homework...

Hey all, I've never taken a formal class on tensor analysis, but I've been trying to learn a few things about it. I was looking at the metric tensor in curvilinear coordinates. This Wikipedia article claims that you can formulate a dot product in curvilinear coordinates through the following...

Hi, I am learning about Polar Coordinates and how they can be written in several equivalent ways. I understand how you can add 360 to angles and use negative angles to represent the same point. However, I have a very hard time understanding how you can write the same point but with a...

1. Given the curves r = 2sin(θ) and r = 2sin(2θ), 0≤θ≤π/2, find the area of the region outside the first curve and inside the second curve 2.not sure which equations to use 3. I got 1 and 1/2 as the area and they were wrong. I do not really know how to work this problem. A...

Homework Statement Sketch the curve r = 1 + 2cosθ in polar coordinates. Homework Equations None that I can think of, it's graphing. The Attempt at a Solution What I was trying to was use the method of finding cartesian coordinates and plugging different values of θ into the equation to...

Homework Statement Plot the Following points(given in polar coordinates). Find all the polar coordinates of each point. a. (2, pi/2) b. (2,0) c. (-2, pi/2) d. (-2,0) Homework Equations none The Attempt at a Solution I have plotted it on a graph but could someone explain to me...

Homework Statement City Latitude(degrees) Longitude(degrees) LOS ANGELES 35.20 N 118.03 W CHICAGO 41.82 N 87.62 W CORVALLIS 44.53 N 123.30 W MONTREAL 45.50 N 73.58 W...

Hi, I need some help understanding the solution to a problem. Equations: x = r.cos(θ) y = r.sin(θ) r = x2 + y2 theta = arctan(y/x)Question: Determine the Jacobian Matrix for (x,y)T and for (r, θ)T SOLUTION: I understand and can compute by myself the Jacobian for (x,y)T, but the solution to...

Homework Statement We have the circle (x - 1)^2 + (y-2)^2 = 1. Find the boundaries of θ and r.Homework Equations x = h + rcosθ y = k + rsinθ The Attempt at a Solution This is a circle with its origin at (1,2) and a radius of 1 so r is between 0 and 1 and the circle lies in the first quadrant...

I'm trying my very best to understand it, but really, I just couldn't get it. I read four books now, and some 6 pdf files and they don't give me a clear cut answer :( Alright, so this integral; ∫e-x2dx from -∞ to ∞, when converted to polar integral, limits become from 0 to 2∏ for the outer...

Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.

Homework Statement do you see how the integral of r is .5? I don't get how that follows?

Homework Statement a elliptic paraboloid is x^2/a^2+y^2/b^2<=(h-z)/h, 0<=z<=h. Its apex occurs at the point (0,0,h). Suppose a>=b. Calculate the volume of that part of the paraboloid that lies above the disc x^2+y^2<=b^2.:confused: 2. The attempt at a solution We normally do the...

I couldn't find any book discussing all of this. =================================================== U+jV=f(x+jy) W=f(z) Ux=Vy Uy= -Vx jWx=Wy <--Cauchy-Riemann equation Uxx+Uyy=0 Vxx+Vyy=0 <--harmonic condition...

Homework Statement Show that ##\iint_{S}(x^2 + y^2)d\sigma = \frac{9\pi}{4}## where ##S = \{(x,y,z): x > 0, y > 0, 3 > z > 0, z^2 = 3(x^2 + y^2)\}## Homework Equations ##\iint_{S}f(x,y,z)d\sigma = \iint_{R}f(r(x,y))\sqrt{[r_x(x,y)]^2 + [r_y(x,y)]^2 + 1}## where ##r : R → ℝ^3, R \in ℝ^2##...

Homework Statement It is given a set defined as: 0≤x≤1, 0≤y≤1-x. With x,y in ℝ. f(x,y)=1 (plane parallel to Oxy plane) They ask you to express the integral ∫∫Setf(x,y)dxdy in polar coordinates and calculate it. Homework Equations x=rcosθ y=rsenθ r=√x2+y2 The Attempt at a...

Homework Statement find the over lapping area of the following equations r=3sin(x) r=1+sin(x)Homework Equations area =1/2 ∫ f(x)^2 dxThe Attempt at a Solution first off I started by finding the intersecting angle by: 3sin(x)=1+sin(x) 2sin(x)=1 sin(x)=1/2 x=pi/6 and the peak is at pi/2 so I...

I am asked to consider the following graph: r2=a+sin(θ), where a=2 I have a picture of this plot, which I have attached, We are asked to find the area of the upper 'cresent' of the curve, contained at the top How would I go about calculating that? I've found that if I plot...

Homework Statement Use polar coordinates to evaluate: ∫sqrt(2)0 ∫sqrt(4-y2)y 1/(1+x2+y2) dxdy Homework Equations The Attempt at a Solution I graphed it and I see r is the part of the elipse sqrt(4-y2) and goes from 0 to ∏/4. I'm not sure how to make the bounds for r or how to...

Homework Statement The lecture notes say that ∇ = urr + (1/r)ur + (1/r2)uθθ. I'm not sure how this comes about. The notes never explain it. Homework Equations (?) The Attempt at a Solution No attempts on the actual homework problem until this ∇ thing is cleared up.

Hi I'm working on area and volume integrals. I was wondering, when you convert to do the integral in polar, cylindrical or spherical co-ordinates, is there a standard set of limits for the theta variable in each case? for example from 0 -pi for polar, 0-2pi for cylindrical? If not how...

Homework Statement Find the area of the shaded region. r=sqrt(θ) Homework Equations A = integral from a to b 1/2r^2dθ The Attempt at a Solution I know how to solve the question, I just don't know what to use for a and b. I tried 0 and 2pi but I am getting the wrong answer...

Homework Statement Find the Exact length of the Polar Curve for r=2(1+cosθ) No limits of Integration were given which I found to be odd. Homework Equations L= ∫√(r^2+(dr/dθ)^2)dθ The Attempt at a Solution r=2(1+cosθ) dr/dθ=-2sinθ L=∫√((2+2cosθ)^2+(-2sinθ)^2)dθ...

Homework Statement Consider the complex number z=(i^201+i^8)/(i^3(1+i)^2). (a) Show that z can be expressed in the Cartesian form 1/2+(1/2)i. (b) Find the modulus of 4z − 2z*. (z* meaning z-bar/complex conjugate of z) (c) Write 2z in polar form. (d) Write 8z^3 in polar exponential form...

Homework Statement Express z=-1+4i in polar for then find z^4 converting to Cartesian form Homework Equations r = sqrt(x^2+y^2) theta = y/x z= r cos (theta) + i r sin (theta) The Attempt at a Solution r= sqrt(-1^2+4^2) = sqrt(17) theta = tan a = 4/1 a = tan^-1...

Homework Statement Im having trouble find the area of integration for this integral which i have to convert to polar: \int_0^2 \int_0^\sqrt{1-(x-1)^2} \frac{x+y}{x^2 + y^2} dydx Homework Equations x = rcosθ y = rsinθ r = x^2 + y^2 The Attempt at a Solution i know exactly what to do to the...

a) I am having a hard time figuring this out. Im saying that x=rcos(\phi) and y = r sin (\phi) where r = a-vt. Im not sure how to work a DE into it. b) I believe it is \omegaa/v since \Delta\Theta= \omegat and \Deltax=vt c) so we have the arc length formula would i use...

Homework Statement The polar ice caps contain about 2.3x10^19 kg of ice. This mass contributes essentially nothing to the Earth's moment of inertia becuase it's located near the poles. Estimate the change in the length of one day if the caps were to melt and redistribute evenly over the...

Can someone please help me on this question. I tried to solve it by integrating 0.5*(1-3sin(θ)^2 from -Pi/2 to 0 but I didnt get the answer.

Homework Statement Find the area enclosed by the inner loop of the curve r=1-3sinθ Homework Equations A=o.5\int r^2 dθ The Attempt at a Solution I found the integral but i don't know how to find the interval at which i will be integrating from. I tried finding when r=0 and it turns...