Vector Definition and 1000 Threads

Homework Statement A long straight wire of radius a and resistance per unit length R carries a constant current I. Find the Poynting vector N = E × H at the surface of the wire and give a sketch showing the directions of the current, the electric field E, the magnetic field H, and N. Integrate...

Homework Statement Consider the surface, S, in the xyz-space with the parametric representation: S: (, ) = [cos() , sin() , ] −1/2 ≤ ≤ 1/2 0 ≤ ≤ os(). The surface is placed in a fluid with the...

In a scientific paper (Neural Networks: A Systematic Introduction, page 86) about the Perceptron Algorithm I found: A good initial heuristic is to start with the average of the positive input vectors minus the average of the negative input vectors. In many cases this yields an initial vector...

Homework Statement At noon two boats P and Q have a position vector (i+7j)km and (3i+8j)km respectively to the origin O. i and j are unit vectors in direction to East and north respectively P is moving south east at 5√2 km/h and Q is moving at a constant velocity of (6i+5j) km/h At time t after...

Homework Statement Homework EquationsThe Attempt at a Solution I took ## \frac { d\vec r}{dt} = 0 ## . This gives ##t_m = \frac { 1} {\sqrt{ 2}} ## ...(1) To calculate ## \alpha ## , ## \cos{ \alpha } = \frac { \vec v \cdot \vec r}{|\vec v||\vec r|} ## ...(2) Then, magnitude of...

Homework Statement Let's say i was given Vector A and B. The Angle between them is 60 degrees. Vector A's magnitude is 40 and Vector B's magnitude is 50. Find magnitude of vector C, if C = vector A - vector B. Homework Equations I'm given the magnitudes, and need to find magnitude of C The...

I'm looking for a diagram or animation that shows the vector potential A (in the form of arrows or whatever) superimposed on the E and B fields of a plane EM wave. Since A is not unique, maybe two or three versions of the diagram (including one with Coulomb guage). An animation with a slider to...

Homework Statement Part C) Please: Homework Equations above,below The Attempt at a Solution so I think I understand the background of these expressions well enough, very briefly, changing the manifold from ## R^n ## to a cylindrical one- ##R^{(n-1)}^{+1}## we need to cater for winding...

Homework Statement A rotating magnetic dipole is built by two oscillating magnetic dipole moments, one along the y-axis and one along the x-axis. Find the vector potential at a point: (0, 0, ##z_0##) along the z-axis. Then find the magnetic field at ##z_0## . As the magnetic field is a function...

It makes me wonder... wikipedia says about a basis: In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.[1] So what is the...

Homework Statement Three close-packed planes of atoms are stacked to form fcc lattice. The stacking sequence of the three planes can be altered to form the hexagonal close packed structure by sliding the third plane by the vector r over the second. If the planes in the fcc structure are all...

Homework Statement The velocity of a solid object rotating about an axis is a field \bar{v} (x,y,z) Show that \bar{\bigtriangledown }\times \bar{v} = 2\,\bar{\omega }, where \bar{\omega } is the angular velocity. Homework Equations 3. The Attempt at a Solution [/B] I tried to use the...

Homework Statement Example 2:[/B] Homework Equations Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y) The Attempt at a Solution Why do my attempt is wrong? The example is using the foundational formula while I use the stock formula from the book, why is there a negative...

Homework Statement The stream function Ψ(x,y) = Asin(πnx)*sin(πmy) where m and n are consitive integers and A is a constant, describes circular flow in the region R = {(x,y): 0≤x≤1, 0≤y≤1 }. Graph several streamlines with A=10 and m=n=1 and describe the flow. Explain why the flow is confined to...

Homework Statement [/B] I would like to ask for Q5b function G & H Homework Equations answer: G: -2pi H: 0 by drawing the vector field The Attempt at a Solution the solution is like: by drawing the vector field, vector field of function G is always tangential to the circle in clockwise...

Mentor note: Thread moved from homework sections as being a better fit in the math technical section. Multiplying components of both sides are also off limits. I am trying to derive vector identities on introduction chapters in various EMT books. For example : (AXB).(CXD) = (A.C)(B.D) -...

In general relativity we demand that the physical law can be stated as a form which does not depend on the choose of particular coordinate system, So the vector field is defined as a changing object following a regular pattern under the transformation of coordinates. For example, we can define...

Homework Statement I am stuck on finding ##W^u## Homework Equations [/B] I have computed the christoffel symbols via comparing the Euler-Lagrange equations to the form expected from geodesic equation. geodesic equation: ##\ddot{x^a}+\Gamma^a_{bc}\dot{x^b}\dot{x^c}=0## covariantly constant...

Homework Statement Homework Equations above The Attempt at a Solution I'm totally stuck as to how to get to started , how to use the hint to decompose a vector given a preferred spacetime direction. All I know is that any tensor can be decomposed into a anti symmetric and a symmetric...

I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.

Hello, I'm currently doing vectors in class, but I don't know what these parts mean. Especially φ(a,b). I understand cos(φ), but I don't understand cos(φ(a,b)). φ is the angle between two vectors a and b, but what does the (a,b) part add? I tried to google, but I couldn't really find...

My answer was [ (1,2), (0,-1) ] I just wanted to make sure with you guys that you could essentially have an infinite amount of answers. For example: [ (1,2), (0,1) ] [ (1,2), (0, -5) ] etc would all suffice, right?

Homework Statement I have been given a changing magnetic field in cylindrical coordinates. The equation is: \begin{equation} B(r,\phi,z) = - \frac {B_1} {2} r \hat{r} + (B_0 + B_1z)\hat{z} \end{equation} I need to be able to find the magnetic field as a function of x, y, and z. Homework...

I'm reading a textbook on electromagnetism. It says that for two vector fields ##\textbf{F}(\textbf{r})## and ##\textbf{G}(\textbf{r})## their inner product is defined as ##(\textbf{F},\textbf{G}) = \int \textbf{F}^{*}\cdot \textbf{G} \thinspace d^3\textbf{r}## And that if ##\textbf{F}## is...

I am trying to find the most efficient way to select points on a 2d plane from a set that maximizes the area of the of the shape they define when joined together. The points are all paired (sharing the same A->B vector), with these pairs also appearing mirrored about the origin. Here is an...

I need some guidance regarding the directional derivative ... Two books I am reading introduce the directional derivative somewhat differently ... these books are as follows: Theodore Shifrin: Multivariable Mathematics and Susan Jane Colley: Vector Calculus (Second Edition)Colley...

Homework Statement Find the vector form of the equation of the line in ℝ2 that passes through P=(2, -1) and is parallel to the line with general equation 2x-3y=1. Homework Equations None. The Attempt at a Solution vector form: x=p+td general form: ax+by=c --------------------------------...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... I need help in fully understanding Browder's comments on Definition 8.9 ... ... Definition...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ... I need help in proving Proposition 8.12 ... ... Proposition 8.12 and the definitions...

I am trying to reconcile apparent differences between the definitions of the derivative of a vector-valued function f: U \rightarrow \mathbb{R}^n (where U \subset \mathbb{R}^m ) of a vector variable from two textbooks ... The textbooks are as follows: Andrew Browder: "Mathematical Analysis...

In this example (https://imgur.com/a/pTFIt) could i define F_1,2 in terms of r-hat_2,1 instead of r-hat_1,2?

Homework Statement Homework Equations The Attempt at a Solution does Vpa has the same direction as Vp? i thought Vp is tangent to the circular path that point P creates( perpendicular to vector Rpa) but from the figure, Vp doesn't seem to be tangent to the path, but Vpa does...

So the Laplacian of a scalar is divergence of the gradient of a scalar field, and it comes out to the double derivative of the field in X, Y, and Z. My book says the Laplacian of a vector field is the double derivative of the X component of the field with respect to X, the double derivative of...

Homework Statement A lightly damped oscillator (mass, m, and spring constant,s, damping constant, b, with one end fixed) under external harmonic force (F0cosωt).(i.e., Standard mass-spring system with lubricated damper, and driven by an external harmonic force.) Since it is lightly...

This is probably fairly simple, but I have a hard time to grasp the concept of the vector potential A in electromgnetism. Especially, in the following equation for the electric field : \vec{E} = - \nabla V - \frac{\partial \vec{A}}{\partial t} When does the second term is not 0 ?

Homework Statement The position vector of an object of mass 0.10 kg at time t in seconds is given by =(^3+5)−4+2^2 Find the velocity and the acceleration as a function of t Homework Equations =(^3+5)−4+2^2 The Attempt at a Solution For velocity I think the equation needs to be...

Homework Statement A physics student studying gravity on the surface of Mars, throws a ball into the air at time t=0s. The ball follows a parabolic trajectory as shown and reaches a maximum height height at t=2s. At t=1s the velocity of the ball is v(t=1s)= 2.00 m/s i + 3.72 m/s j where i and j...

Homework Statement i know how to find the basis of a subspace of R2 or R3 but I can't figure out how to find the basis of a subspace of something like R2,2. I even have an example in my book which i managed to follow nearly till the end but not quite... Given matrix: A= 6 -9 4 -4 show that...

Homework Statement [/B] F =< 2x, e^y + z cos y,sin y > (a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path. (b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...

Homework Statement Find the unit vectors along which the given functions below increase and decrease most rapidly at P0 . Then find the derivatives of the functions in these directions. Homework Equations solution: The Attempt at a Solution why are the derivatives’ values along these...

I have a sample problem : Photon, with energy E, is absorbed by m. Then, mass m becomes m' (i think it's relativistic mass). Calculate the new mass. My teacher solved the problem as follows; P1=( E/c, P3dphoton); // momentum of the photon P2=(mc, 0); // momentum of the stationary mass...

Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...

Homework Statement Suppose we have a Electic field, E (vector) = kr2r(vector). Find the total charge contained in the sphere of radius R centered in the origin. solution- here E can also be written as kr3r∧, where r∧ is the unit vector this is the given question , so obiously the best way to...

Homework Statement Find th distance of point P to the plane OQR, O being the origin. "PS" is perpendicular to plane "OQR" P(3,-2,-1) S(x,y,z) O(0,0,0) Q(1,3,4) R(2,1,-2) Homework Equations A) The equation in terms of (x,y,z) obtain by triple product of OS,OQ & OR. OS.(OQXOR)=0... [all being...

\begin{align*}\displaystyle \vec{V_2} &=\int_0^3 \left[\left( \frac{4}{\sqrt{1+t}}\right){I}-\left(7t^2 \right){j} +\left(\frac{14t}{\left(1+t^2 \right)^2}\right){k}\right] dt \\ &=\left[\int_0^3 \frac{4}{\sqrt{1+t}} dt \right] {I} -\left[\int_0^3 7t^2 dt\right] {j} +\left[...

Is this the correct partial derivative of B? ##\vec{B} = \frac{g \vec{r}}{4 \pi r^3}## ##\frac{\partial \vec B}{\partial r}## = ##-3\frac{g \vec{r}}{4 \pi r^4} + \frac{g}{4 \pi r^3 }(\frac{\partial r_r \hat r}{\partial r})##

1. Problem: Consider vector field A##\left( \vec r \right) = \frac {\vec n} {(r^2+a^2)}## representing the electric field of a point charge, however, regularized by adding a in the denominator. Here ##\vec n = \frac {\vec r} r##. Calculate the divergence of this vector field. Show that in the...

Homework Statement Which of the following correctly describes the velocity vector in each case? 2. The attempt at a solution I got it wrong at first. My new attempt (I have a sneaking suspicion that I am missing something important): For the first picture: dtheta_1/dt<0 - the angle is...

Hello all, I have a theoretical question regarding subspaces. If V is a subset of a vector space, and we wish to show that V is a vector space itself, we need to show 3 things. Some references say we need to show: a) V is not empty b) V is closed under + c) V is closed under scalar...

Greetings, can somebody show me how to calculate such a term? P= X E² where X is a third order tensor and E and P are 3 dimensional vectors. Since the result is supposed to be a vector, the square over E is not meant to be the scalar product. But the tensor product of E with itself yields a...