Articles of vector analysis

What exactly are pseudovectors and pseudoscalars? And where could I read about them?

I can’t find good information on the internet. In my mathematical physics class the definition of a vector was given as: That object with magnitude and direction which doesn’t change under inversions. Pseudovectors do change. Scalars are that magnitude that don’t change with inversions. Pseudoscalars do change. Inversions were loosely defined as inverting all components […]

Calculate the surface integral $\iint_S (\nabla \times F)\cdot dS$ over a part of a sphere

How can I calculate the integral $$\iint_S (\nabla \times F)\cdot dS$$ where $S$ is the part of the surface of the sphere $x^2+y^2+z^2=1$ and $x+y+z\ge 1$, $F=(y-z, z-x, x-y)$. I calculated that $\nabla\times F=(-2 , -2 ,-2)$. It’s difficult for me to find the section between the sphere and the plane. Also, I can’t calculate […]

Dimensions analysis in Differential equation

Differential equation of solitary wave oscillons is defined by, $$ \Delta S -S +S^3=0 $$ How can we write this equation as, \begin{equation} \langle(\vec{\nabla}S)^2\rangle+\langle S^2\rangle-\langle S^4\rangle=0 \tag{1} \end{equation} where $\langle f\rangle:=\int d^Dx f(x)$. Furthermore, another virial identity can be found from the scaling transformation ($\vec{x}\to \mu \vec{x}$) by extremizing the scaled ($\vec{x}\to\mu \vec{x}$) of the […]

Rate of change of direction of vector-valued function

This should be easy, but I cannot figure out what I’m doing wrong, and it’s killing me. Let $\mathbf{f}(t)$ be a function from $\mathbb{R}$ to $\mathbb{R}^3$. I want to find the “rate of change of the angle between $\mathbf{f}(t)$ and nearby vectors (from $\mathbf{f}$)”. I’m going to assume $|\mathbf{f}(t)|=1$, because it makes the writing easier, […]

Two 2d vector angle clockwise predicate

I’ve got three vectors v1[0,1], v2[-1,1], v3[1,1]. I calculated the angles beetween v1-v2 and v1-v3 using the forumla dot(vec1, vec2) / ||vec1|| * ||vec2|| and the both angles are the same (45 degrees). How to determine that the vector v2 lies on the countercloclwise side (CCW) form v1 (CW angle is 335 degress), and v3 […]

Calculate the viewing-angle on a square (3d-calc)

I’m in big trouble: My program (Java) successfully recognised a square drawn on a paper (by its 4 edges). Now I need to calculate, under which angle the webcam is facing this square. So I get the 4 coordinates of the shape, and I already had an idea: You could have a look on the […]

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a diagonal matrix whose diagonal elements are the elements of $x$ and $\mathbf{1}$ is a vector whose elements are equal to 1. I will already be very happy to find a […]

$z=f\left(xy,\, \frac y x \right)$ Differentiate

I have a problem solving this problem, since I find it difficult to find the derivatives of $z$ with respect to $u$ and $v$, I would appreciate any help you can give me. $$ z= f\left(xy,\ \frac y x \right) $$ Where x and y belong to $R$ And $$xy=u$$ $$\frac{y}{x}=v$$ Prove that $$x^2\frac{\partial^2 z}{\partial […]

The Velocities of the Contact Points of Two Rolling Curves are Equal at the Instant of Contact

Consider two 2-Dimensional rigid bodies surrounded by two planar smooth curves. Suppose that the two bodies are in the same plane and in contact with each other such that they are rolling with respect to each other. To demonstrate the meaning of rolling, suppose that the points $C_1$ and $C_2$ in the figure below are […]

Product rule for scalar-vector product

Let $\mathbf F : \mathbb R^p \to \mathbb R^s$ and $\phi : \mathbb R^p \to \mathbb R$ be differentiable functions. Let the function $\mathbf G$ be defined as follows: $$\mathbf G : \mathbb R^p \to \mathbb R^s \qquad \mathbf G(\mathbf y) = \phi(\mathbf y)\mathbf F(\mathbf y)$$ Furthermore, let $y_0$ be a point in $\mathbb R^p$. […]