📘 vector calculus

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 $$\vec{X}(t) = \cos t \hat{e}_1 + \sin t \hat{e}_2 + t^3 \hat{e}_3$$

1. Stating the problem: (a) Determine $A\times B$ and angle $\theta$ between the two vectors where $A = 3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$ and $B = 2\mathbf{i} + 3\mathbf{j}

1. **State the problem:** We want to compute the surface integral $$\iint_{S} \text{curl} \, \mathbf{F} \cdot d\mathbf{S}$$ where $$\mathbf{F}(x,y,z) = -y^{2} \mathbf{i} + x \mathb

1. **Problem (a):** Find the cross product $\mathbf{A} \times \mathbf{B}$ and the angle $\theta$ between vectors $\mathbf{A} = 3\mathbf{i} - 2\mathbf{j} + 4\mathbf{k}$ and $\mathbf

1. **State the problem:** We need to evaluate the line integral $$\int_C \mathbf{F} \cdot d\mathbf{r}$$ where $$\mathbf{F}(x,y) = \left(8 - 14xy^2 + 2y e^{2x}\right) \mathbf{i} + \

1. **State the problem:** We are given a vector field $$\mathbf{F}(x,y,z) = (5 \ln y - yz) \mathbf{i} + \left(\frac{5x}{y} - xz\right) \mathbf{j} + (-xy - \pi) \mathbf{k}$$.

1. **Stating the problem:** We are given the equation:

1. The problem is to understand the meaning and calculation of the operator \nabla \times, known as the curl of a vector field. 2. The curl of a vector field \( \mathbf{F} = (F_x,