What is Vector: Definition and 1000 Discussions

The notation I think best describes it is ## F = \lVert\int^{space}_s|\vec{V}|ds\rVert ## So you have a vector field V in a 3d space. For each point you integrate over all of space (similar to a gravitational or electromagnetic field) *but* vectors in opposite directions do not cancel, they...

I could try to apply the Liénard-WIechert equations immediatally, but i am not sure if i understand it appropriately, so i tried to find by myself, and would like to know if you agree with me. When the information arrives in ##P##, the particle will be at ##r##, such that this condition need to...

I'm following 《A First Course In General Relativity》.On page 72,it says"If the surface is spacelike,the outward normal vector points outwards.If the surface is timelike,however,the outward normal vector points inwards"I wonder why and how?

Suppose I have some interaction potential, u(r), between two repelling particles. We will name them particles 1 and 2. I want to find the force vectors F_12 and F_21. Would I be correct in saying that the x-component of F_12 would be given by -du/dx, y-component -du/dy etc? And to find the...

My interest is on part ##a## only. Is the markscheme correct? I have ##BA= -b+a= a-b.## It therefore follows that ##|a-b|## is the Length of ##BA##...or are we saying it does not matter even if we have ... ##AB.## Cheers

My interest is only on part (b), For part (i), My approach is as follows, ##PB=PO+OB## ##PB=-\dfrac{2}{5}a +b## ##AD=AO+OD## ##AD=-a+\dfrac{5}{2}b## Therefore, ##PB=\dfrac{2}{5}AD## For part (ii), we shall have; ##QD=QB+BD## ##QD=\dfrac{2}{7}AB+\dfrac{3}{2}b##...

I will only care about the ##t## and ##x## coordinates so that ##(t, z, x, x_i) \rightarrow (t,x)##. The normal vector is given by, ##n^\mu = g^{\mu\nu} \partial_\nu S ## How do I calculate ##n^\mu## in terms of ##U## given that the surface is written in terms of ##t## and ##x##? Also, after...

So, we have A, the magnetic vector potential, and its divergence is the Lorenz gauge condition. I want to solve for the two vector fields of F and G, and I'm wondering how I should begin##\nabla \cdot \mathbf{F}=-\nabla \cdot\frac{\partial}{\partial t}\mathbf{A} =-\frac{\partial}{\partial...

(An even longer-winded version was written and deleted out of mercy.) Assume an AC voltage at zero degrees applied to an ideal parallel RLC circuit. For a predominantly inductive circuit, the vector diagram for current should show the supply current in the fourth quadrant (i.e. with lagging...

I am passing through some difficulties to understand the reasoning to derive the electric potential of an oscilating dipole used by Griffths at his Electrodynamics book: Knowing that ##t_o = t - r/c##, What exactly he has used here to go from the first term after "and hence" to the second term...

Hi. I have the Marsden an Tromba vector calculus book 6th edition. I was wondering which software was used to create the books graphs. I attach two graphs as an example. Thanks

For the case that there is only a potential ##\sim 1/r##, I have already proven that the time derivative of the Lenz vector is zero. However, I'm not sure how I would "integrate" this perturbation potential/force into the definition of the Lenz vector (as it is directly defined in terms of the...

Hi PF! I have a function ##f(s,\theta) = r(s,\theta)\hat r + t(s,\theta)\hat \theta + z(s,\theta)\hat z##. How can I plot such a thing in Mathematica? Surely there's an easier way than decomposing ##\hat r, \hat \theta## into their ##\hat x,\hat y## components and then using ParametricPlot3D?

[Moderator's note: Spin off from another thread due to topic change.] I was thinking about the following: can we take as a basis vector a null (i.e. lightlike) vector to write down the metric ? Call ##v## such a vector and add to it 3 linear independent vectors. We get a basis for the tangent...

I think all the options are wrong. Since I = Δp = m1v1' - m1v1, I draw it like this: Is my drawing wrong? Thanks

Does the electric field vector takes into account the field's radial direction? Usually when we calculate the electric field, we use ##\vec E = \frac{kq}{r^2}\vec j##, which is a straight line vector of a positive charge q's electric field. This electric field points from a positive charge q to...

I need to use hermiticity and electromagnetic gauge invariance to determine the constraints on the constants. Through hermiticity, i found that the coefficients need to be real. However, I am not sure how gauge invariance would come into the picture to give further contraints. I think the...

In many books it is just written that ##\Delta(\frac{1}{r})=0##. However it is only the case when ##r \neq 0##. In general case ##\Delta(\frac{1}{r})=-4\pi \delta(\vec{r})##. What abot ##\mbox{div}(\frac{\vec{r}}{r^3})##? What is that in case where we include also point ##0##?

Hi, I was thinking about the following problem, but I couldn't think of any conclusive reasons to support my idea. Question: Let us imagine that we have two vectors ## \vec{a} ## and ## \vec{b} ## and they point in similar directions, such that the inner-product is evaluated to be a +ve...

Known: 1) The mass of the ball is ##m## (constant ##\frac{dm}{dt} = 0##) 2) ##v(0) = v_{0}## 3) Air drag force magnitude ##| \vec F_{D} | = B \cdot | \vec v(t) |## (##B \in R##) 4) The ramp is frictionless. 5) The magnitude of Earth's acceleration = ##g## I'm not sure if θ is known or not, and...

Hi there, I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove : Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E## Then...

I'd like a good set of notes or a textbook recommendation on how to approach vector differential equations. I'm looking for something that isn't specific to one type of application like E&M, fluid dynamics, etc., but draws heavily from those and other fields for examples. I'd strongly prefer a...

In a problem of an oscillating electric dipole, under appropriate conditions, one can find, for the potential vector calculated at the point ##\vec{r}##, the expression ##\vec{A}=\hat{k}\frac{\mu_0I_0d}{4\pi}\frac{cos(\omega(t-r/c))}{r}## where: ##\hat{k}## is the direction of the ##z-axis##...

I have seen two expansions of a vector potential, $$\mathbf A=\sum_\sigma \int \frac{d^3k}{(16 \pi^3 |\mathbf k|)^{1/2}} [\epsilon_\sigma(\mathbf k) \alpha_\sigma (\mathbf k) e^{i \mathbf k \cdot \mathbf x}+c.c.],$$ and $$\mathbf A=\sum_\sigma \int \frac{d^3k}{ (2 \pi)^3(2 |\mathbf k|)^{1/2}}...

I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...

Question: Equation: Attempt: Can someone verify my answer?

Question: Equations: My attempt: Could someone confirm my answer please?

Using the equations mentioned under this question, I came up with following analysis and directions of velocities on either side of ##x_1##. Also, I'm not sure if there is an easier qualitative way to know the velocity directions rather than do a detailed Calculus based analysis?

Anyone know of an online course or set of video lectures on John Hubbard's textbook on Vector Calculus, Linear Algebra, and Differential Forms?

My answer so far in |S| = √3 /2 *hbar but the question states it must be an angular momentum. Is this an angular momentum or am I missing something? Thanks

Not sure how to show that because ##\vec{v} = |v|\hat{v} = 3|e|\hat{e}##, but since ##\vec{e}## is a unit vector we know ##|e| = 1## so our equation now becomes ##\hat{v} = \frac{3\hat{e}}{|v|}##. So, we're left to the task of showing that ##|v| = 3## in order to conclude that ##\hat{v} =...

My question is on part (c) only. Find the markscheme solution below; Mythoughts on this; (Alternative Method) i used the simultaneous equation ##λ####\left[\dfrac {1}{2}a -\dfrac {1}{4}b\right]##=##\left[ -\dfrac {3}{4}b+ ka\right]## where ##OR=k OA## ##- \dfrac {1}{4}bλ##=## -\dfrac...

What would the correct form of this? $$\stackrel{\leftrightarrow}{A} \cdot \vec{b} = \stackrel{\leftrightarrow}{C} : \vec{d}$$ I'd like to know which one is correct form 1.) $$ \vec{b} = (\stackrel{\leftrightarrow}{A})^{-1} ( \stackrel{\leftrightarrow}{C} : \vec{d} ) $$ 2.) $$ \vec{b} =...

I've calculated the intensity for every point charge which are EA = 6.741 x 10¹³ NC¯¹ EB = 4.494 x 10¹¹ NC¯¹ EC = 6.741 x 10¹³ NC¯¹ and I am pretty sure about this far but I am struggling to calculate the X-axis intensity and Y-axis intensity to find the entire approximate intensity with the...

I need a justification that ##|\dfrac {dr}{dt}|##=##\dfrac {ds}{dt}## cheers guys... all the other steps are easy and clear to me...

I looked at this question and i wanted to ask if we could also use; ##C## =## c_2 ##(##-\dfrac {3}{2}i## +## j - 3k)## ... cheers This problem can also be solved by using the approach of cross product ##A×B##...

In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation) ## \nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j} ## I'm trying to prove that this covariant...

I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.

This is the question, Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...

The curl is defined using Cartersian coordinates as \begin{equation} \nabla\times A = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}. \end{equation} However, what are the...

I have doubts about the wording of the exercise: (1) energy density is ##u=\varepsilon_0 (cB)^2## but since the question asks for mean energy density should I perhaps average over ##cos^2 (\omega t)## (there due to the ##B^2##) and thus use ##<u>=\frac{1}{2}\varepsilon_0 (cB)^2##? (2) it seems...

I understand that angular velocity is technically not a vector so does that mean the cross product of the radius vector and the angular velocity vector, the tangential vector, is also not a vector?

Consider the following \begin{equation} \nabla\phi=\nabla\times \vec{A}. \end{equation} Is it possible to find ##\vec{A}## from the above equation and if so, how does one go about doing so? [Moderator's note: moved from a homework forum.]

According to equation (2.33) divergence of the Poynting vector or the outflow of electromagnetic power is equal to the stored magnetic field, stored electric field and ohmic losses. My contradiction is the following: Inside a steady state DC current carrying wire, there will presumably be a...

According to e.g. Keith Conrad (https://kconrad.math.uconn.edu/blurbs/ choose Complexification) If W is a vector in the vector space R2, then the complexification of R2, labelled R2(c), is a vector space W⊕W, elements of which are pairs (W,W) that satisfy the multiplication rule for complex...

Hi, I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##). As far as I understand it, the vector...

Divergence formula $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$ If we express it in terms of the components of ##\vec{A}## in unit basis using $$A^{*j} = \sqrt{g^{jj}} A^{j}$$ , we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}}...

Hey! 😊 Construct a class named Vector that expresses the meaning of the vector of numbers. The initialization function, __init__ will take as a argument a tuple corresponding to the vector. This tuple will be stored in the variable named coord. Also, __init__ will calculate the Euclidean...

There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...