Vector Definition and 1000 Threads

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand the vector space of alternating multilinear functions ... The relevant text from Shifrin reads as follows: In...

Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...

We commonly have E and B defined as: But how can I work in electric field E, instead of vector potential A?

Hello everybody. The Lagrangian for a massive vector field is: $$\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{m^2}{2}A_\mu A^\mu$$ The equation of motion is ##\partial_\mu F^{\mu\nu}+m^2A^\nu = 0## Expanding the EOM with the definition of ##F^{\mu\nu}## the Klein-Gordon equation for...

Hi, I'm trying to find all the valid surfaces that go through a vector field so that the normal of the surface at any point is equal with the vector from the vector field at the same point. The vector field is defined by the function: $$ \hat N(p) = \hat L(p) \cos \theta + \hat R(p)...

The difference between light and very heavy atoms reflects itself in these two schemes. My question is why one scheme for the vector sum is necessarily the right & suitable sum model for one case, and the 2nd scheme suits the 2nd case ? In other words, why & how the relative magnitude of the...

Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...

I don't understand, please ckeck Let V=\Bbb{R}^2 and {u=(u_1,u_2), v=(v_1.v_2)}\in\Bbb{R}^2 , {k}\in \Bbb{R} define of operation u\oplus v = (u_1+v_1,u_2+v_2) and k \odot u =(2ku_1,2ku_2) check V is vector over field \Bbb{R} ? ________________________________________________________________ I...

Homework Statement The diagram shows parallelogram ABCD. (you don't really need the diagram) vector AB= (2 above, 7 below) and vector AC= (10 above, 11 below) The point B has coordinates (5, 8) (a) Work out the coordinates of the point C. The point E has coordinates (63, 211) (b) Use a...

I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus). If you also have online recommendations please feel free to write them.

I'm trying to figure out this volume integral, a triple integral, of a 9-variable function. 3 Cartesian-dimension variables, and 6 primed and un-primed co-ordinates. After the volume integration, the un-primed co-ordinates will have been gotten rid of, leaving a field function in terms of...

I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a non-uniform (monotonically decreasing) distribution of angles. In math speek: Ai = uniform(0,2π) dA = Ai - Aj dA is not uniform. Here is a rough image of...

In a recent thread, the following was posted regarding the "no hair" theorem for black holes: In the arxiv paper linked to, it says the following (p. 2, after Theorem 1.1): "Hawking has shown that in addition to the original, stationary, Killing field, which has to be tangent to the event...

Given the following vectors: how can i determine that Θ = Δp/p ? I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...

In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$. But from the transformation of contravariant vector...

In my pre-calculus textbook, the problem states: A 200-pound cart sits on a ramp inclined at 30 degrees. What force is required to keep the cart from rolling down the ramp? The gravitational force can be represented by the vector F=0i-200j In order to find the force we need to project vector...

Hey! :o Let $S_{X,3}$ be the vector space of cubic spline functions on $[-1,1]$ in respect to the points $$X=\left \{x_0=-1, x_1=-\frac{1}{2}, x_2=0, x_3=\frac{1}{2}, x_4\right \}$$ I want to check if the function $$f(x)=\left ||x|^3-\left |x+\frac{1}{3}\right |^3\right |$$ is in $S_{X,3}$...

Let us have some localized density of sources, S, in a plane, each of which produces a localized circular vector field. Let us work in polar coordinates. Let the density of sources, S = Aexp(-r^2/a^2) and let each source have circular vector field whose strength is given by exp(-(r-r_i)^2/b^2)...

Homework Statement Homework Equations v = I + j + k v = d/t The Attempt at a Solution I thought the answer was as simple as: v = 63i + 0j + 0k, since the car only has motion in one direction... ...but I got it wrong, so clearly I'm missing something here.

I use the ##(-,+,+,+)## signature. In the Schwarzschild solution $$ds^2=-\left(1-\frac{2m}{r}\right)dt^2+\left(1-\frac{2m}{r}\right)^{-1}dr^2+r^2d\Omega^2$$ with coordinates $$(t,r,\theta,\phi)$$ the timelike Killing vector $$K^a=\delta^a_0=\partial_0=(1,0,0,0)$$ has a norm squared of...

Homework Statement Calculate |u+v+w|, knowing that u, v, and w are vectors in space such that |u|=√2, |v|=√3, u is perpendicular to v, w=u×v. Homework Equations |w|=|u×v|=|u|*|v|*sinΘ The Attempt at a Solution [/B] Θ=90° |w|=(√2)*(√3)*sin(90°)=√(6) Then I tried to use u={√2,0,0}...

Homework Statement The position vector of a particle changes: Only by its module. Only by its direction. What can be said about the trayectory of the movement of the particle? Obtain the answer analitically. Homework Equations None. The Attempt at a Solution I think that the trayectory...

this is what is given so by addition $$\begin{bmatrix}x_1\\y_1\\5z_1\end{bmatrix} \oplus \begin{bmatrix} x_2\\y_2\\5z_2 \end{bmatrix} = \begin{bmatrix} x_1+x_2\\y_1+y_2\\5z_1+5z_2 \end{bmatrix} = \begin{bmatrix} X\\Y\\10Z \end{bmatrix}$$ uhmmmm really?

On the set of vectors $\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}\in \Bbb{R}^2 $ with $x_1 \in \Bbb{R}$, and $y_1$ in $\Bbb{R}^{+}$ (meaning $y_1 >0$) define an addition by $$\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} \oplus \begin{bmatrix} x_2 \\ y_2 \end{bmatrix} = \begin{bmatrix} x_1 + x_2 \\...

Homework Statement a) A point charge + q is placed at the origin. By explicitly calculating the relevant line integral, determine how much external work must be done to bring another point charge + q from infinity to the point r2= aŷ ? Consider the difference between external work and work...

Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of...

So say our inner product is defined as ##\int_a^b f^*(x)g(x) dx##, which is pretty standard. For some operator ##\hat A##, do we then have ## \langle \hat A ψ | \hat A ψ \rangle = \langle ψ | \hat A ^* \hat A | ψ \rangle = \int_a^b ψ^*(x) \hat A ^* \hat A ψ(x) dx##? This seems counter-intuitive...

Good day everybody, I'm currently working on the Grover algorithm. You can also illustrate this process geometrically and that's exactly what I have a question for. In my literary literature one obtains a uniform superposition by applying the Hadamard transformation to N-qubits. So far that's...

Homework Statement You wish to row straight across a 63 meter-wide river. You can row at a steady 1.3 m/s relative to the water and the river flows at 0.57 m/s. In what direction should you head, and how long would it take you to cross the river? Homework EquationsThe Attempt at a Solution...

Homework Statement A 0.54 kg block of ice is sliding by you on a very slippery floor at 2.1 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0034 seconds. The block eventually slides at an angle of 21 degrees from its original...

Please see the attached page to see what I'm talking about. In the top right paragraph, it states to use "the positive direction of the x-axis". It is given that ##θ_2=30°## and it shown visually at the bottom of the page. In the problem it's using -60° and I'm not sure how they're getting that.

Dear all, I am trying to find if these two sets are vector subspaces of R^3. \[V=\left \{ (x,y,z)\in R^{3}|(x-y)^{2}+z^{2}=0 \right \}\] \[W=\left \{ (x,y,z)\in R^{3}|(x+1)^{2}=x^{2}+1 \right \}\] In both cases the zero vector is in the set, therefore I just need to prove closure to addition...

In 2-D Cartesian coordinate system let's there exist a scaler field Φ(x1,x2) ,now we want to find how Φ changes with a curve which is described by the parameter(arc length) s dΦ/ds=(∂Φ/∂xi)dxi/ds Can we say for Cartesian coordinate system that along the curve at any s dxi always points in the...

Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!

I'm trying to use LaTeX to graph both the vectors of the electric field around a dipole and the field lines. So far I have a quiver plot of the vector field: I obtained this by using the code \begin{tikzpicture} \def \U{(x-1)/((x-1)^2+y^2)^(3/2) - (x+1)/((x+1)^2+y^2)^(3/2)} \def...

Homework Statement I am trying to show that Majorana vector current vanishes. I am following this article and I am trying to get to the very right hand side of eq. (27). Homework Equations \psi_M^C = \psi_M,\\ \psi^C_M = C \overline{\psi}_M^T,\\ C^T=-C, \hspace{1cm} C^T\gamma_{\mu}C =...

I want to visualize the concept of divergence of a vector field.I also have searched the web.Some says it is 1.the amount of flux per unit volume in a region around some point 2.Divergence of vector quantity indicates how much the vector spreads out from the certain point.(is a...

Could there be a connection between Robert Zimmermann's work (McMaster Univ. Toronto) on Vector Plasma, and Jenkins and Fischbach's (Perdue Univ.) work on variations in the rate of radioactive decay for elements on Earth in relation to solar activity? Only looking for a confirmation that their...

I am learning the basics of differential geometry and I came across tangent vectors. Let's say we have a manifold M and we consider a point p in M. A tangent vector ##X## at p is an element of ##T_pM## and if ##\frac{\partial}{\partial x^ \mu}## is a basis of ##T_pM##, then we can write $$X =...

Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking. Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the...

When comparing Newtonian and GR views of gravity, I came across a vector expression in the Newtonian form which happens to integrate to the total potential energy of a system of masses, even in the case of dynamic situations: ##-\mathbf{x}\cdot\rho \, \mathbf{g}##, where ##\mathbf{x}## is...

Hello all! I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is. It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove...

Homework Statement A vector n of magnitude 8 units is inclined to x,y and z axis at 45, 60 and 60 degrees resoectively.If the plane passes through (root2, -1, 1) and is normal to n then find its equation. Homework Equations (r-a).n=0 where r is position vector of a point on plane, a is a point...

The theorem is as follows: All finite dimensional vector spaces of the same dimension are isomorphic Attempt: If T is a linear map defined as : T : V →W : dim(V) = dim(W) = x < ∞ & V,W are vector spaces It would be sufficient to prove T is a bijective linear map: let W := {wi}ni like wise let...

I'm given equations of surfaces and asked for the vector function that represents the intersection of the two surfaces. For ex: $$x^2 + y^2 = 4$$ and $$z = xy$$ In the solutions manual the answer is given like this: a sum of terms of cos t and sin t (is this polar form?). The way I did wasn't...

I seem to have come across a new problem I am trying to programm a sloping pipe system to match an array of vector points. I thought I had it all sorted out until I tested with a very high slope angle. With a normal slope of around 2% everything looks fine. If I increase the slope to 30% there...

I know how to find the cos(theta) between two vectors but I do not know how to find the sin(theta). vector w=i+3j vector v=<5, 2>

I am reading Miroslav Lovric's book: Vector Calculus ... and am currently focused n Section 1.3: The Dot Product ... I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows: It...

I have a vector ##\textbf{v} \in \mathbb{R}^{3N}## and a function ##\textbf{Ψ} : \mathbb{R}^{3N} \longrightarrow \mathbb{R}^p## such that ##\textbf{Ψ}(\textbf{v})=0##. Why the set ##T=\{ \textbf{x} \in \mathbb{R}^{3N} \ | \ \textbf{Ψ}(\textbf{x})=0 \}## has dimension ##n=3N-p##?