Vector Definition and 1000 Threads

Suppose we have n vector fields ## Y_{\left(1\right)},\ldots,Y_{\left(n\right)} ## such that at every point of the manifold they form a basis for the tangent space at that point . I have to prove that: $$\frac{\partial Y_\mu^{\left(\sigma\right)}}{\partial x^\nu}-\frac{\partial...

My book says, If the position vectors a, b, c of three points A,B,C and the scalars α, β, γ are such that αa + βb + γc=0,and α + β + γ = 0, then the three points A,B,C are collinear. On the other hand, If the position vectors a, b, c, d of the four points A,B,C,D (no three of which are...

Hi I have problems with the following task I now wanted to try to calculate the vector potential, which according to my professor's script is defined as follows: $$\mathbf{A}(\mathbf{x}) = \frac{1}{c} \int d^3\mathbf{x}' \frac{\mathbf{j}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|}$$ I have...

My idea is as follows. Each mode of the quark field is characterized by a wave vector k. Each wave vector corresponds to a point in k-space. This set of points representing different modes forms a manifold. Each point in k-space can be assigned a three-dimensional vector space that represents...

We know that a = vdv/dx But is it applicable in only one dimensional components or is this actually a vector equation? If so then how do we exactly differentiate with respect to position 'vector'?

Here is the solution: From vector identity, $$\nabla (\vec A \times \vec B) = \vec B \cdot \nabla \times A - \vec A \cdot \nabla \times \vec B $$ If ##\vec B = 0##, then $$\nabla \vec S = \nabla \frac{\vec E \times \vec B}{\mu_0} = \frac{1}{\mu_0} (\nabla \times E \cdot \vec B - \cdot E \cdot...

I posted a follow-up question on an earlier question, but did not get a response. So I will start a new thread. A vector ##\vec V## and a covector/one-form ##\tilde V## represent the same object, right? ##\vec V=V^i \vec e_i = V_i \vec e^i = \tilde V##. We also have ##\mathbf g(\vec V)=\tilde...

Assume s is a set such that Fs denotes the set of functions from S-->F where F is a field such as R, C or [0,1] etc. One requirement for F to be a vector space of these functions is closure- e.g. that sums of these functions are in the space: For f,g in Fs the sum f+g must be in Fs hence...

I greet everyone I am faced with such a question First 100m to the right, then 300m down, then 150 left diagonally, even I don't think this is exactly right, I think there will be 150x sin30 or cos30, but I'm not sure, I need to do 200xcos60 afterwards, but I couldn't settle the question. can...

Source: https://tutorial.math.lamar.edu/Problems/CalcIII/SurfIntVectorField.aspx Alright so my confusion lies in the following step: Consider the side x = 0. Okay, from the formula (I am just going to insert an image here) Okay, so gradient of f would just be <1, 0, 0>, which is simply i...

Given the following equation: R = ((Q-P) / |Q-P|) ⋅ V where Q, P, and V are 3 dimensional vectors, R is a scalar, "⋅" denotes the dot product, and |Q-P| is the magnitude of Q-P. Assuming Q, V, and R are known and given 3 independent equations with different values for Q, V, and R that...

Hello! I have the following vector ##\mathbf{A} = R(-\sin(\omega t)\mathbf{x}+\cos(\omega t)\mathbf{y})##, where ##\mathbf{x}## and ##\mathbf{y}## are orthogonal unit vectors (aslo orthogonal to ##\mathbf{z}##). I want to calculate ##\nabla \times \mathbf{A}##, but I am a bit confused. The curl...

Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

This question has been bugging me for quite a while, That what do we mean by direction of angular velocity or torque. As we know that the direction of angular velocity or torque even is determined by right hand thumb rule, and it come out to be perpendicular to the rotational plane. So my...

I can imagine x + y = 1 to be line in xy - plane but how can x + 2y + z = 3 be a line, not a plane? Thanks

This post parallels a post I made in electrical engineering regarding the S plane. I thought I would post an equivalent in basic physics. So, given a graph of velocity vs time we have on the vertical axis meters/sec and the hormonal axis just meters. Given a plot of V vs t we know the area...

In the S plane we have a real component, usually called sigma, and the imaginary component, jw, in radians/sec. The real component is sometimes called nepers per second, with nepers being dimensionless. However, if we draw a vector in the s-plane, say s - s1, in polar form, what are the units...

Is this correct? Ignore my bad drawings

V is a vector space and W is a subset of V. could W be a vector space but not a subspace of V?

Some axially symmetric star has two independent KVFs, ##T## and ##\Phi##. We don't know the expressions for these at all points -- the only thing we know is that as ##r^2 + z^2 \rightarrow \infty##, that ##T \rightarrow \partial/\partial t## and ##\Phi \rightarrow \partial/\partial \phi##. The...

This is the general suggested approach given in a textbook. My question is why can I not directly write it in vector form? E1 vector + E2 vector =0 should be valid no? Why are they choosing to write E1 mag + E2 mag=0 Then find a vector form Then convert the magnitude equation into a vector...

The problem and solution is, However, I am confused how they get ##\vec a = (1, 2)## (I convert from column vector to coordinate form of vector). I got ##\vec a = (a_1, a_2) = (a_1, 2a_1) = a_1(1, 2)## however, why did they eliminate the constant ##a_1##? Thanks for any help!

O level question; i used similarity would appreciate an easier approach for 2 marks. The ms solution (approach) is not clear to me. Here it is; My approach; using similarity Any insight welcome its a 2 mark question- cannot seem to find easier way though i suspect reflection.

Hello, I have watched a really good Youtube video on linear algebra by Dr. Trefor Bazett and it made me think about a question... () Personal Review A basis in 2D space is formed by any two independent vectors that are not collinear geometrically. Any vector in the 2D space can then be...

In Dirac's GTR. Sec. 12 (p. 22), he wants to show the equivalence of: (a) Vanishing of the curvature tensor ##R^\beta_{\sigma\nu\rho}=0##; or equivalently, the equality of mixed second covariant derivatives ##A_{\nu:\sigma:\rho}=A_{\nu:\rho:\sigma}##. (b) Path independence of parallel transport...

I first calculated the time using y = (viy)(t) + 0.5gt^2 where y is the vertical displacement which is 0 for the ball landing back on the ground, viy is the initial vertical velocity ie 16.55m/s and g = -9.8m/s}^2. I get 2 values for t, t=0 and t= 3.377s. Then using the equation x = (vix)(t) =...

Going through this ( Revision) A salways your insights are quite helpful. I would like to go through all these questions; i will start with (5), ##\left( \dfrac {x} {y} \right)## = ##\left( \dfrac {10 \cos 40^0} {10 \sin 40^0} \right)## + ##\left( \dfrac {4 \cos 150^0} {4\sin 150^0}...

Ok. My problem is what angle to choose when adding vector. Statement does not tell me which one is the "first" force vector. So, when using the law of sine formula I get two results. First, using cosine to get the magnitude: $$\vec c = \sqrt{a^2 + b^2 +2ab\cos\theta},$$ $$\vec c = \sqrt{15^2 +...

How do you derive the position vector in a general local basis? For example, in spherical coordinates, it's ##\vec r =r \hat {\mathbf e_r}##, not an expression that involves that involves the vectors ## {\hat {\mathbf e_{\theta}}}## and ## \hat {{\mathbf e_{\phi}}}##. But how would you show this?

Can a vector subspace have the same dimension as the space it is part of? If so, can such a subspace have a Cartesian equation? if so, can you give an example. Thanks in advance;

All I know is that e subscript r must be a vector cos the book says so, but what does it mean, is it, a konstant in vector form? I'm confused by it (page one, chapter one spacetime and geometry by SeanCaroll) Help is appreciated Edit. Is vector r describing the curvature that takes place ?

I am extremely confused with how to represent vectors that do not start at the origin in spherical coordinate system. If they did start at the origin, the vector to any point is simply ##r\pmb{\hat{r}}##; however, what if it does not start at the origin as in the question above? One thing I can...

What am I trying to do for ##\vec V=\vec a \phi## : ##R.H.S= \oint \vec V \cdot d \vec \lambda=\oint \vec a \phi \cdot d \vec \lambda=\vec a \cdot \oint \phi d \vec \lambda ## ##L.H.S= \iint_S \vec \nabla \times \vec V \cdot \vec d \sigma=\iint_S \vec \nabla \times (\vec a \phi) \cdot \vec d...

So I do know that there does exist a generalization of the cross product (the exterior product), but this question does not concern that (and I would prefer it not ) I know that the cross product (that Theodore Frankel, for example, calls "the most toxic operation in math") works in 3D only...

Hi, I have made the following ContourPlot in mathematica and now I wanted to ##\vec{r}_1= \left(\begin{array}{c} -1 \\ 1 \end{array}\right)##, ##\vec{r}_2= \left(\begin{array}{c} 0 \\ \sqrt{2} \end{array}\right)## and ##\vec{r}_3= \left(\begin{array}{c} 1 \\ 1 \end{array}\right)## insert the...

I am not sure why latex is not rendering, but here is the question. The answer is ##\frac{a^2}{8}## and for the love of my life, I don't know how. Can you please help me with this?

Is angle Vag 180 since the vector is a straight line pointing left? Also you can add 30 degrees with 150 which will be 180?

So i found the magnitude which is (-1)^2 + (-2)^2 = P^2 = Sqrt(5) Then I used the inverse tan function to find the angle (direction) theta = arctan (-2/-1) = 63.8 degrees Im confused with my 63.8 degrees since the angle in the graph looks greater than 63.4 degrees I subtracted 180 by 63.8 and...

So for 1 I know it's Yes you can, but I don't understand what uniquely means here so I can't say if it's uniquely or not. for 2 I've never seen a 2-D vector broken into 3 reference axies so I guess No? What really confuse me is the answers which goes 1-C and 4-A

I don't get what is the difference when I am asked to re-solve components and find projections to axes other than the Y and X I know that the parallelogram works for the first one and the dot product for the second but what's the diffrence!

The problem is actually a solved example My attention is focused in understanding the displacement vector diagram as it has to deal with question (iv). I have no problem understanding the velocity vector sketch below: As you can see ND is the velocity of the river current due East and NE is...

This homework statement comes from a research paper that was published in SPIE Optical Engineering. The integral $$\int\int_{-\infty}^{\infty}drdr'W(\vec{r})W(\vec{r'}) \vec{r} \cdot \vec{r'}=0$$ is an assumtion that is made via the following statement from the paper : "Since...

I am looking at this now; pretty straightforward as long as you are conversant with the formula: anyway i think there is a mistake on highlighted i.e Ought to be ##-\dfrac{15}{37}(i+6j)## just need a confirmation as at times i may miss to see something. If indeed its a mistake then its time...

Suppose a stationary reference frame, any other reference frame will have a clock that advances at t'= t/gand is moving away at velocity v. g is the relativistic factor. we can write: C d t' = Cdt/g and d x = v dt this can be written (Cdt',Cdx)=(C/g,C v) dt ; since 1/g2+ v2 =...

Given that c = 3i + 4j and d = i - 2j, find u if uc + d is parallel to i + 3j, this is the question in the solution, it says that we have to multiply the 3 with the i i do know that this is the ”method“ to do this question but I’d like a bit of explanatio. I don’t understand why the 3 is...

I don't understand anything about how 1.18 came to be (Red Rectangle), any help would be greatly appreciated!

An airplane has an air velocity of 500 km/h [N 30 E] and encounters a wind from [S 75 W] at 180 km/h, find the ground velocity. Make sure you draw a big, labelled diagram. Please help! I’m understand the calculations that need to be done (cosine law then sine law for the angle) but I’m a little...

But I don't understand how is that it ends up on the west side if both vectors were pointing east.

At first I thought that this force vector ## \vec F = 3 \hat x + 2 \hat y ## is a function of ## x ## and ## y ##, which is to say that its magnitude and direction vary with the x and y positions, but this is not so, right? It's just a force with a constant magnitude and direction. And I can...