# Tagged Questions

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### is it necessary that curl of 2d vector is perpendicular to the plane.

I am just confused, help me guys. The question comes up, because we say that curl is either clockwise or anti-clockwise at a point.
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### Proof: Force always perpendicular and motion in a plane implies that the trajectory is a circle

I am looking for a proof for a physics problem. Consider a particle which is subject to a force $\vec{F}(t)$ with $|\vec{F}(t)| = \text{const}$ which is always perpendicular to the velocity ...
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### Helmholtz decomposition - motivation

Our lecturer presented us the Helmholtz decomposition of smooth vector fields. He added a proof, but he didn't provide any single motivation - e.g. where Helmholtz used the decomposition or for which ...
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### How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...
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### Finding the magnitude of a vector product between two vectors?

Vector $\overrightarrow{A}$ has magnitude $11.0m$ and vector $\overrightarrow{B}$ has magnitude $16.0m$ . The scalar product $\overrightarrow{A}\bullet \overrightarrow{B}$ is $79.0m^2$. What is the ...
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### Physics Vector Problem - Airplane

Heres the question: A plane leaves the airport in Galisto and flies $140$km at $68.0^∘$ east of north and then changes direction to fly $255$km at $48.0^∘$ south of east, after which it makes an ...
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### How to determine the magnitude of a resultant vector?

I was able to determine a resultant vector based on the sum of two vectors and told to express them in vector units. Here is my answer that was correct: ...
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### How would you prove $E = -\vec{\nabla} V$ from the electric potential's line integral?

Basically, we were given an equation: $$V_1 - V_2 = \int_{r_1}^{r_2}\vec{E}\cdot d\vec{r}$$ where $\vec{E}$ is the electric field distribution and $d\vec{r}$ is the displacement vector of the charge. ...
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### Tensor Projection

I'm currently reading "Vector and Tensor Analysis with Applications" by A.I. Borisenko and I.E. Tarapov, and I'm having trouble following a particular mathematical step in where the author projects ...
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I am starting with the standard definition of an angular momentum operator in quantum mechanics given as $$\mathbf{L} = k(\mathbf{r}\times\mathbf{p}) = k(\mathbf{r}\times\nabla),$$ where ...
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### Trouble understanding a common vector calculus example

I have difficulty understanding the following vector calculus example. Text can be found here. It is the 5th Q&A -- starting with equation (31.1035).It concerns finding the vector potential of a ...
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### Let $F(x,y,z) = -c(r/||r||^3)$ be the force resulting from the inverse square law…

$c$ is a constant and $r = (x,y,z)$. Show that $\displaystyle f(x,y,z) = \frac{c}{\sqrt{x^2+y^2+z^2}}$ is a potential function for $F$. What can be concluded from any path from point $A$ to point $B$ ...
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### Parametric curve of intersection - line integral with respect to arc length

This comes from Apostol's Calculus, Vol. II, Section 10.9 #14: A uniform wire has the shape of that portion of the curve of intersecion of the two surfaces $x^2+y^2=z^2$ and $y^2=x$ connecting the ...
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### Determine the flow and amplitude equation for thermal energy (with Del operator)

It is a question vector calculus and Maxwell's laws. I put it this way. Let's say, we are working in a $3$-Dimensional space ( e.g $x\cdot y\cdot z = 4\cdot3\cdot2$, a certain room/class of that size ...
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I learned electrodynamics. According to the vector potential determination, $$\mathbf B = [\nabla \times \mathbf A ],$$ Coulomb gauge, $$\nabla \mathbf A = 0,$$ and one of Maxwell's equations, $$... 2answers 2k views ### Finding the component of a vector tangent to a circle Problem Given a vector and a circle in a plane, I'm trying to find the component of the vector that is tangent to the circle. The location of the tangent vector is unimportant; I only need to know ... 1answer 122 views ### Help with volume integration I need help solving this integral (\hat{z} denotes the polar axis):$$\int_V\dfrac{\vec{r}\cdot(\vec{r}-c\hat{z})}{|\vec{r}|^3|\vec{r}-c\hat{z}|^3} dV$$Where V denotes all space. Attempt:$$2\pi ...
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Here's a question from my homework: Two vectors are given by a = 1.9i + 3.2j and b = 1.6i + 8.6j. Find (a)|a × b|, (b)a · b, (c)(a + b) · b, and (d) the component of a along the direction of ...