📘 vector calculus

1. **Problem Statement:** Given the space curve defined by $$x = \cos t, \quad y = \sin t, \quad z = 3t,$$ show that the curvature $$\kappa = \frac{1}{10}$$ and the torsion $$\tau

1. The problem is to understand and sketch the vector field given by $$\mathbf{F}(x,y) = \left\langle -\frac{y}{2}, \frac{x}{2} \right\rangle.$$\n\n2. This vector field assigns to

1. **Problem statement:** Given the vector equation $\mathbf{v} = \mathbf{w} \times \mathbf{r}$, where $\mathbf{w}$ is a constant vector and $\mathbf{r}$ is the position vector, pr

1. **Problem statement:** Given the vector equation $\mathbf{v} = \mathbf{w} \times \mathbf{r}$, where $\mathbf{w}$ is a constant vector and $\mathbf{r}$ is the position vector, pr

1. **State the problem:** We need to find the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F} = x^2 \mathbf{i} - yz \mathbf{j} + x \cos z \mathbf{k}$ and the

1. **Problem Statement:** Verify Green's theorem for the given line integrals and curves, and explain its application in mesh processing.

1. **Problem statement:** Verify Green's theorem for the line integral $$\oint_C (2x - y)\,dx + (x + 3y)\,dy$$ where $C$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(0,1)$

1. **Problem:** Show that $$\nabla \times \left( \frac{\mathbf{a} \times \mathbf{r}}{|\mathbf{r}|^6} \right) = \frac{6 (\mathbf{a} \cdot \mathbf{r})}{|\mathbf{r}|^8} \mathbf{r} - \

1. The problem asks to determine the sign (positive, negative, or zero) of the line integral of given vector fields along specified oriented paths. 2. Recall that the line integral

1. **Problem Statement:** We want to understand how the equation for the height $h$ of a parallelepiped, given by $h = a \cdot \frac{b \times c}{|b \times c|}$, is derived. 2. **Ba

1. The problem describes a semicircle drawn on a horizontal line segment with its diameter coinciding with the segment. 2. The semicircle is centered on the horizontal line, and an

1. **Énoncé du problème :** Calculer le flux du champ de vecteurs $\vec{V} = (xz, z, -\frac{z^2}{2})$ à travers la surface $S$ définie par $z = x^2 + y^2$ avec $z \leq 1$, orientée

1. نبدأ بكتابة متجه السرعة المعطى: $$\mathbf{V} = (6tx + z^2 y) \mathbf{i} + (3t + xy^2) \mathbf{j} + (xy - 2xyz - 6tz) \mathbf{k}$$

1. **Problem Statement:** Show that the vector field $\vec{F} = (x^2 + x y^2) \vec{i} + (y^2 + x^2 y) \vec{j}$ is irrotational and find its scalar potential.

1. **Problem Statement:** Find the curl of the vector field $\mathbf{A} = 2xz^2 \mathbf{i} - yz \mathbf{j} + 3xz^3 \mathbf{k}$. 2. **Recall the formula for curl:**

1. The problem asks to identify the condition for a vector \( \mathbf{V} \) to be irrotational. 2. By definition, a vector field \( \mathbf{V} \) is irrotational if its curl is zer

1. Сформулюємо задачу: потрібно знайти циркуляцію векторного поля $\vec{a} = (2y - z)\vec{i} + (x + y)\vec{j} + x\vec{k}$ по контуру $L$, який є перетином площини $x + 2y + 2z = 4$

1. **Постановка задачі:** Знайти циркуляцію векторного поля $$\vec{a} = (2y - z) \vec{i} + (x + y) \vec{j} + x \vec{k}$$ по контуру $$L$$, який є перетином площини $$x + 2y + z = 4

1. **Problem Statement:** Verify Green's theorem for the line integral \(\oint_C (5x^4 - 8y^2)\,dx + (4y - 6x^3)\,dy\), where \(C\) is the boundary of the region defined by \(x=0\)

1. **Problem Statement:** Given the scalar function $$\Phi(x,y,z) = xy^2 + y^2z$$, find at the point $(1,1,-1)$: i) The gradient $$\nabla \Phi$$

1. **State the problem:** Given the scalar function $$Q(x,y,z) = xy^2 + y^2z$$, we need to find: