Cartesian Definition and 561 Threads

please someone explain me the following expression for Cartesian unit vectors expressed by the cylindrical unit vectors: http://web.mit.edu/8.02t/www/materials/modules/ReviewB.pdf at page B-8 line B.2.4 i would like to know which steps led to it. thanks, Chen

Homework Statement An alpha particle (the nucleus of a helium atom) is at rest at the origin of a Cartesian coordinate system. A proton is moving with a velocity of v towards the alpha particle in the xˆ direction. If the proton is initially far enough away to have no potential energy, how...

Homework Statement A cannon shoots a ball at an angle θ above the horizontal ground. (a) Neglecting air resistance, use Newton's second law to find the ball's position as a function of time. (Use axes with x measured horizontally and y vertically.) (b) Let r(t) denote the ball's distance...

Homework Statement For the vectors in a Triangle, with a = 16, b = 12, and c = 20 what are (a) the magnitude and (b) the direction of A x B (c) the magnitude and (d) the direction of A x C (e) the magnitude and (f) the direction B x C this is Vector Multiplication. Homework Equations...

Suppose T=T(r,θ)=G(x,y) How do you prove ∇T(r,θ)=∇G(x,y)? I can think of some arguments in favor of this equality, but I want an actual proof or a very good intuitive argument. My arguments in favor go something like this: -Gradient vectors should be the same because if my directional...

Definition/Summary Cartesian coordinates are ordinary rectangular coordinates in a flat Euclidean space. Cartesian form of a complex number is the form x + iy, where x and y are real. Cartesian product of two or more sets is the most general product set, the direct product, with the symbol...

Hello. I think that you know an experiment called Cartesian diver. I can use some heads of matches like Castesian divers like there: I know how it works. However, I don't know why the heats of matches don't go down in some moment, but first one, then second with bigger force, then third with...

Does anyone have a proof of what a isotropic cartesian tensor should look like in three or four dimensions?

So I'm wondering, should I use Cartesian or Polar Coordinates to store intergalactic objects in DB? I'm currently prototyping a game idea that can be oversimplified as a spaceship simulator in infinite space. I'm considering grouping objects together so that they have a "parent super-space"...

Homework Statement determine the Cartesian equation of the plane through the points (3,0,1) and (0,1,-1) and perpendicular to the plane with equation x-y-z+1 = 0 Homework Equations The Attempt at a Solution Well I know the normal of the plane (a,b,c) dotted with (1,-1,-1) = 0...

I'm not sure if it is the right place to post my questions to. please let me know if this should be somewhere else. 1. let each of the vectors $A=5a_x-a_y+3a_z$ $B=-2a_x+2_ay+4a_z$ $C=3a_y-4a_z$ extend outward from the origin of the cartesian coordinate system to points A,B, And...

Homework Statement Homework Equations The Attempt at a Solution Is this the correct setup? \int^{\pi}_{\frac{3\pi}{4}}\int^{2\pi}_{0}\int^{\sqrt{2}}_{0}\frac{1}{\rho^{2}} rho^2 Sin\phi d\rho d\theta d\phi I gave up on itex. It was either that or my computer flying out the...

Hi all. I'm having trouble understanding the cartesian product of a (possible infinite) family of sets. Lets say \mathcal{F} = \{A_i\}_{i \in I} is a family of sets. According to wikipedia, the cartesian product of this family is the set \prod_{i \in I} A_i = \{ f : I \to \bigcup_{i...

Homework Statement Convert -2+2√3i to polar coordinates. Homework Equations r = √x2+y2 θ = tan-1(y/x) The Attempt at a Solution I am confused because θ = tan-1(2√3/2) = tan-1(√3) = -π/3 and r = 4, so that would make the polar form 4cis(-π/3), but the calculator gives: 4cis(2π/3). I...

Cartesian product of indexed family of sets The definition of a Cartesian product of an indexed family of sets (X_i)_{i\in I} is \Pi_{i\in I}X_i=\left\{f:I \rightarrow \bigcup_{i \in I} \right\} So if I understand correctly, it's a function that maps every index i to an element f(i) such...

I'm running through the following problem in an EM text of mine: Calculate the flux of a point charge Q at the origin through a cube centered through the origin by both direct integration and Gauss's Law (essentially prove Gauss's Law for a point charge). Now I got the answer of Q/ε0 via...

Homework Statement For every pair of sets (A,B) we have P(AxB)=P(A)xP(B) Prove or disprove the above statement. Homework Equations The Attempt at a Solution I have attempted solving this using A={1,2} and B={a,b} AxB={(1,a),(1,b),(2,a),(2,b)}...

mathematical "connection" in the cartesian product What is the mathematical connection between elements of a cartesian product ##A\times{}B## and the elements of the sets ##A## and ##B##? In other words, what is the difference between the set ##A\times{}B## and just any set ##Z## with...

Hello I've got a problem with Cartesian Geometry and cannot find a solution. A will appretiate any help I can get. b) Show that [AQ] has equation cx + by = -2ac c) Prove that the third median [BR] passes through the point of intersection G of medians [OP] and [AQ] Cheers!

Homework Statement Determine the magnitude of the projected component of F along AC. Express this component as a Cartesian vector.Homework Equations Information I'm working with. The Attempt at a Solution Most of my attempts have proven fruitless. I understand that in order to find a...

Hello, I am trying to build a high-speed X/Y cartesian robot which vacuums up defected parts from a moving conveyor. The end effector of the robot has a venturi vacuum integrated. Because the vacuum is constant, I need a Z-axis to go up/down for the end effector to get the defects otherwise...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

I was trying to study vectorial kinematics in all its fullness, without decorating formulas, only deducting all vectorially through mathematical definitions. I felt much difficulty, because it's a puzzle of many pieces and I not found a embracing explanation in any book. Someone could explain to...

I am revising vectors at the moment. So, this is not a homework question. In 2-D cartesian space, I have two vectors, A = <a,2> and B = <1,3>. I want to find the limit of the angle between the vectors as a -> -∞. Geometrically, I know that direction of vector B approaches 180° as a→-∞...

Homework Statement ∫∫√(x^2+y^2)dxdy with 0<=y<=1 and -SQRT(y-y^2)<=x<=0 Homework Equations x=rcos(theta) y=rsin(theta) The Attempt at a Solution 0.5<=r=1, we get r=0.5 from -SQRT(y-y^2)<=x by completing the square on the LHS then, 0<=theta<=pi But, when I calculated the...

Suppose I was asked if G \cong H \times G/H . At first I considered a familiar group, G = S_3 with its subgroup H = A_3 . I know that the quotient group is the cosets of H, but then I realized that I have no idea how to interpret a Cartesian product of any type of set with elements that aren't...

I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors $$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$ $$\hat{j} =...

Homework Statement I have a problem that is the curl of jln(rsinθ) Since this is in spherical, why is there a bold j in the problem? Doesn't that indicate it's a unit vector in cartesian coordinates? Can I do the curl in spherical coordinates when I have a cartesian unit vector in the...

another question: convert $|\frac{1-i}{3}|$ to polar form i am getting $\frac{\sqrt{2}}{3} e^{\frac{i\pi}{4}}$ but the solutions say: $e^{\frac{-i\pi}{4}}$ i did $ x = r\cos(\theta)$ and $y=r\sin(\theta)$ so $\frac{1}{3} = {\frac{\sqrt{2}}{3}}\cos(\theta)$ $\frac{1}{3} = \cos(\theta)$ And...

Write a short FORTRAN90 subroutine to convert Cartesian coordinates (x, y, z) to spherical polar coordinates (r, q, f) using • Write a FORTRAN90 program which uses this subroutine to convert the following (x, y, z) coordinates which are read from a text file and stored within a single vector...

Homework Statement Consider the following Lagrangian in Cartesian coordinates: L(x, y, x', y') = 12 (x^ 2 + y^2) -sqrt(x^2 + y^2) (a) Write the Lagrange equations of motion, and show that x = cos(t); y =sin(t) is a solution. (b) Changing from Cartesian to polar coordinates, x = r...

Homework Statement See attached image Homework Equations The Attempt at a Solution For the first half of the question, ordered pairs would be (1, [1]), since 1 and [1] are the multiplicative identities in these rings. but no matter how many times we add (1, [1]) to itself...

Homework Statement I need to answer a bunch of topological questions based on the cartesian product of two sets, but I'm not entirely sure how to graph them out. I have A = [1,2)U{3} and B = {1, (1/2), (1/3), ...}U[-2,-1). S = A x B, and I need the graph of S. Could anyone help me with...

Homework Statement Compute the line integral of \vec{v} = (rcos^{2}\theta)\widehat{r} - (rcos\theta sin\theta)\widehat{\theta} + 3r\widehat{\phi} over the line from (0,1,0) to (0,1,2) (in Cartesian coordinates) The Attempt at a Solution Well, I expressed the path as a...

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...

i have a problem : A small loop antenna in free space and centered about the origin on the xy-plane is producing a (far-field) radiation electric field (in phasor notation) : http://postimg.org/image/63tm76h5l/ and their solution : http://postimg.org/image/6mdm6roh9/ i don't understand how...

Hello, I would like to rotate the Cartesian coordinate system ( i=(1,0,0); j=(0,1,0); k=(0,0,1) ) so that angles between new and the old axes be equal to α, β and γ, respectively. Is any simple way similar to the Euler transformations to accomplish that?

I've have two questions, but if my assumption is incorrect for the first, it will also be incorrect for the second. (in-terms of physics.) For a two dimensional cylinder, using cylindrical co-ordinates (as follows), taking v(subscript-r) => velocity normal to cylinder surface & v(subscript-phi)...

Homework Statement In Cartesian coordinaate system, we describe the rotation of a cylinder. The axis of the cylinder has the same direction as the basis vector e3. Angular velocity is described by vector w = 2e1 - 5e2 + 7e3 rad/s. I must find the velocity vector (v) of a point P that is...

okay first hello. i hope someone can help me with this. i am working on a mystery geocache that the hint says is a cartesian problem. i will copy the info below and add a link to the page if you want to look at it. again thanks for any help...

Homework Statement For cable AD it is known that the magnitude is 14 kips, x-component has a value of -6.216, the direction angle in the z-direction is 83.63°, and Fy is less than zero. Find forces in Cartesian vector form, coordinates of point D if it lies on the x-z plane and point A is (0...

Hello, Just as a warning before anyone reads my question I am not a mathematician, just an engineer with moderate math skills he wants to expand. So I'm writing some engineering software which involves defining/interation/modification of geometry within a cartesian system but I currently...

I am having a really tough time trying to solve this particular problem I can find alpha easily 139 degrees which is correct. However when I go to try and solve for beta or gamma I get this huge mess and I am unable to solve for either one, Please help me I know I am overlooking something but I...

Homework Statement I have a vector field (which happens to be a magnetic field) H = -\frac{I }{2 \pi r}u\varphi u\varphi is the unit vector which is in the cylindrical coordinate system with only the \varphi component nonzero so it circles around the z-axis. r is the radius of the circle...

1. The cartesian coordinates of a point in the xy plane are x=−9.92 m, y=−2.8 m.Find the distance "r" from the point to the origin.Answer in units of m Homework Equations A=sqrt(Axi^2 + Ayj^2) The Attempt at a Solution i did the following: A=sqrt((-9.92m)^2 + (-2.8m)^2) and got...

Let's say i have a parametric equation: x = t^2 y = t^3 + 4t Even though this is a 2nd and 3rd degree parametric equation, i can isolate and express in terms of y = f(x) because the parametric equation for x involves only one term for t. Thus: t = sqrt(x) and y = sqrt(x)^3 + 4(sqrt(x))...

If I had a sphere centred at the origin with x > 0, y > 0 and z > 0 Would the angle boundaries be: 0 < θ < pi/2 0 < α < pi/2 ?

How to translate r = 2 /(2 - cos(theta)) to cartesian coordinates: so far: r = 2 /(2 - cos(theta)) r = 2 /(2 - cos(theta)) |* (2 - cos(theta)) both sides r (2 - cos(theta))= 2 2*r - rcos(theta) = 2 | know x = rcos(theta) 2*r - x...

The cartesian product ∏X = Xi of a countable family {Xi} of regular spaces is zero-dimensional i f and only i f all spaces Xi , are zero-dimensional. I wonder if the countability assumption is just to ensure the regularity of the product space ,or it is crucial for the clopen basis. Thank's

Homework Statement The position vectors of two points a and b on a line are < 2,1,7> and < 1,4,-1> Find the parametric vector equation for any point on the line. Find the linear Cartesian equation of the line using x, y, z as coordinates. Homework Equations For the parametric eq. x...