📘 vector calculus

1. **State the problem:** We are given a vector function $$r(t) = \langle 3\sqrt{2}t, e^{3t}, e^{-3t} \rangle$$ and asked to find the unit tangent vector $$T(t)$$, the unit normal

1. **Problem Statement:** (a)(i) State Stokes' theorem without proof.

1. **Problem Statement:** (a)(i) State the Divergence theorem without proof.

1. **State the problem:** Calculate the outward flux of the vector field $F(x,y,z) = x^2 \mathbf{i} + y^2 \mathbf{j} + z^2 \mathbf{k}$ across the closed surface $S$ which is the bo

1. **Problem Statement:** State the Divergence theorem without proof. 2. **Divergence Theorem Formula:**

1. **State the problem:** We want to find a non-zero vector function $u$ satisfying the equation $$\nabla \times u - f u = 0,$$ where $f$ and $g$ are given functions, $g$ is not co

1. **Problem statement:** Given a scalar function $f$ in $\mathbb{R}^3$ such that $|\nabla f|$ depends only on $f$, prove that there exist functions $g$ and $h$ such that $(f,g,h)$

1. **Stating the problem:** We want to understand Green's Theorem geometrically. 2. **The theorem:** Green's Theorem relates a line integral around a simple closed curve $C$ to a d

1. Let's start by understanding what vector calculus is. Vector calculus deals with vectors, which are quantities that have both magnitude and direction. 2. A common problem in vec

1. The problem asks to find the value of $a$ so that the parameter curve $\vec{s}(t) = \begin{pmatrix} t^2 - 5t + a \\ t + 1 \end{pmatrix}$ passes through the point $P(0,2)$. 2. To

1. **مسئله:** مشتق تابع برداری $$\vec{R}(t)$$ داده شده است: $$\vec{R}'(t) = e^t \sin t \vec{i} + \sqrt{2} e^t \sin t \vec{j} + \vec{k}$$

1. مسئله را بیان می‌کنیم: می‌خواهیم انتگرال خطی $$\int_C \vec{F} \cdot d\vec{r}$$ را روی منحنی پارامتری داده شده محاسبه کنیم.

1. **Problem statement:** Given a vector function $\mathbf{v}(t)$ and a constant vector $\mathbf{k}$, find the time derivatives of: (i) $|\mathbf{v}|^2$

1. **Problem statement:** Given a vector function $\mathbf{v}(t)$ and a constant vector $\mathbf{k}$, find the time derivatives of: (i) $[\mathbf{v}]^2$

1. **Problem statement:** Calculate the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \) where \( S \) is the triangle with vertices \( (1,0,0), (0,2,0), (0,1,1) \) and

1. **State the problem:** We need to understand and graph the parametric vector function $$r(t) = (-\sin t, -\cos t, 2t)$$ for $$t$$ in the interval $$[-\frac{3\pi}{2}, \frac{3\pi}

1. **State the problem:** We need to find the work done by the gradient field $\vec{F} = \langle yz, xz, xy \rangle$ when moving from point $A = (3,1,4)$ to point $B = (1,5,9)$.\n\

1. **State the problem:** We need to solve the integral $$\int \left(\int (F_x + G_y)^2 \right)$$ where $F_x$ and $G_y$ denote partial derivatives of functions $F$ and $G$ with res

1. **Problem:** Evaluate the surface integral $$\iint_S \mathbf{F} \cdot \mathbf{n} \, ds$$ where $$\mathbf{F} = (3x - 2z)\mathbf{i} - (2x + y)\mathbf{j} + (y^2 + 2z)\mathbf{k}$$ a

1. **State the problem:** We want to evaluate the surface integral $$\iint_S \mathbf{F} \cdot d\mathbf{S}$$ where $$\mathbf{F}(x,y,z) = (2x^2 + e^y)\mathbf{i} + 5xy\mathbf{j} + 8z\

1. **State the problem:** Find the equation of the principal normal line and the osculating plane to the curve $$\vec{X}(t) = \cos t \hat{e}_1 + \sin t \hat{e}_2 + t^3 \hat{e}_3$$