📘 vector calculus

1. **State the problem:** We want to show that $$\nabla \times \left( \frac{\vec{a} \times \vec{x}}{r^3} \right) = -\frac{\vec{a}}{r^3} + 3 \frac{(\vec{a} \cdot \vec{x}) \vec{x}}{r

1. **State the problem:** Given the vector function $\vec{F} = (-\sin(2t))\vec{i} + \cos(t)\vec{j} + 2t\vec{k}$, find: (i) The magnitude of the first derivative $\left|\frac{d\vec{

1. Stating the problem: Given the vector function $$\mathbf{F} = (-3x^2 y - x^4)\mathbf{i} + (e^x y + 3 \sin x \cdot y)\mathbf{j} + (-2 x^2 \cos y)\mathbf{k},$$ we need to find the

1. **State the problem:** Find the unit tangent vector to the curve $$\mathbf{r}(t) = k t \ln(t) \mathbf{\hat{k}} + 4 t^3 \mathbf{\hat{i}} - 2 t \mathbf{\hat{j}}$$ at $$t=2$$. 2. *

1. We are asked to find the unit tangent vector to the curve $$\mathbf{r}(t) = k t \ln(t) \mathbf{i} + 4 t^3 \mathbf{j} - 2 t \mathbf{k}$$

1. Statement of the problem: Given the time-dependent vectors $A(t)=2t^3\mathbf{i}+t^4\mathbf{j}+2t\mathbf{k}$ and $B(t)=4t^6\mathbf{i}-1\mathbf{j}-4t\mathbf{k}$, compute $\dfrac{d

1. The Green's Theorem states that for a positively oriented, piecewise-smooth, simple closed curve $C$ in the plane and a region $D$ bounded by $C$, if $P(x,y)$ and $Q(x,y)$ have

1. The problem asks to verify the vector calculus identity $$\mathbf{r} \times (\mathbf{r} \times \mathbf{v}) = \mathbf{r}(\mathbf{r} \cdot \mathbf{v}) - (\mathbf{r} \cdot \mathbf{

1. **State the problem:** Given the vector field $v = (3xyz, 2xy, -xyz)$ and scalar field $\phi = 3x^2 - yz$, find (i) $\text{div } v$, (ii) $v \cdot \nabla \phi$, and (iii) $\text

1. **Problem statement:** Given the vector function $$\vec{r}(t) = (2\sin 3t, t, 2\cos 3t)$$ find at the point $$P(0, \pi, -2)$$: a) The equations of the tangent, normal, and binor

1. **State the problem:** Given the vector field $$\mathbf{F} = \left( \frac{y}{\sqrt{1-x^2 y^2}} + 2xy^3 + 6 \right) \mathbf{i} + \left( \frac{x}{\sqrt{1-x^2 y^2}} + 3x^2 y^2 + 7

1. We are given the vector field $F(x,y,z) = yi + zj + xk$ and the parameterizations $x = t$, $y = t^2$, and $z = t^3$. 2. Substitute the parameterizations into the vector field:

1. Énonçons le problème : On a un vecteur $\mathbf{U} = (xy, y^3z, xz)$ et on doit calculer le rotationnel $\nabla \times \mathbf{U}$ puis appliquer la divergence $\nabla \cdot (\n