📘 vector calculus

1. **Problem statement:** Evaluate the line integral $$\int \mathbf{A} \cdot d\mathbf{r}$$ where $$\mathbf{A} = xy^2 \mathbf{i} + x^2 y^2 \mathbf{j}$$ along two paths: i) The strai

1. **State the problem:** We need to evaluate the line integral $$\int \vec{A} \cdot d\vec{r}$$ where $$\vec{A} = (5xy - 6x^2) \hat{i} + (2y - 4x) \hat{j}$$ along the curve $$y = x

1. **State the problem:** We are given a vector field \( \vec{F} = (6xy + z^3)\mathbf{i} + 3x^2(-z)\mathbf{j} + 3xz^2(-y)\mathbf{k} \). We need to prove that \( \vec{F} \) is irrot

1. **Problem:** Find the divergence of the vector field \(\vec{F} = x^{2} y \vec{i} - (z^{3} - 3x) \vec{j} + 4 y^{2} \vec{k}\). 2. **Formula:** The divergence of a vector field \(\

1. **Problem:** Match the vector equation $\mathbf{r}(t)$ with its corresponding graph. 2. **Understanding vector equations:** Each vector equation describes a curve or line in 3D

1. The symbol \nabla (called "nabla" or "del") is a vector differential operator used in vector calculus. 2. It is defined as \nabla = \left(\frac{\partial}{\partial x}, \frac{\par

1. **Problem:** Given the vector function $r(t) = t\mathbf{i} + (1 - t)\mathbf{j}$ for $0 \leq t \leq 1$, describe the curve and find its key features. 2. **Formula and rules:** Th

1. **Problem Statement:** Compute the circulation of the heat/data flow vector field \( \vec{F} = (yz^2, xz^2, x^2 + y^2) \) along the boundary of the circular region defined by \(

1. **Problem Statement:** Calculate the circulation of the vector field $\mathbf{F} = (yz + x, xz - y, xy + z)$ around the triangular loop with vertices $A(1,0,0)$, $B(0,1,0)$, and

1. **State the problem:** Determine if the vector field $$\vec{F} = \left( 4y^{2} + \frac{3x^{2}y}{z^{2}} \right) \vec{i} + \left( 8xy + \frac{x^{3}}{z^{2}} \right) \vec{j} + \left

1. **State the problem:** Determine if the vector field \( \mathbf{F}(x,y,z) = (4y + 3xyz)\mathbf{i} + (8xy + xz)\mathbf{j} + (11 - 2xyz)\mathbf{k} \) is conservative. 2. **Recall

1. Statement of the problem. We need to derive the identity

1. **State the problem:** We need to find the work done by the force field $\mathbf{F} = (y^2, 2xy)$ in moving a particle along the curve $y = x^2$ from the point $(0,0)$ to $(1,1)

1. The problem involves understanding key concepts in vector calculus including parametrization of curves, arc length, line integrals, vector fields, and related theorems. 2. Param

1. **Problem:** Show that $\operatorname{div} \left( \frac{\mathbf{r}}{|\mathbf{r}|^3} \right) = 0$ where $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$ and $r = |\mathbf{r}| = \

1. **Problem:** Verify Stokes' theorem for the line integral $$\int_C (2x^2 - y^2)\,dx + (x^2 + y^2)\,dy$$ where $C$ is the region bounded by the lines $x=0$, $y=0$, $x=2$, and $y=

1. **Problem Statement:** Evaluate the surface integral $$\iint_S \mathbf{A} \cdot \mathbf{n} \, dS$$ where $$\mathbf{A} = z \mathbf{i} + x \mathbf{j} - 3y^2 z \mathbf{k}$$ and $$S

1. The problem is to determine if the vector function \( \mathbf{v} = \{6xy + z^3, 3x^3 - z, 3xz^2 - y\} \) is irrotational and solenoidal. 2. Recall the definitions:

1. **State the problem:** We are given a vector function $$\mathbf{v} = \{6 Xx Yy + Zz^3, 3 Xx^3 - Zz, 3 Xx Zz^2 - Yy\}$$ and need to determine if it is irrotational or rotational

1. The problem is to determine if the vector field \( \mathbf{v} = \{6xy + z^3, 3x^3 - z, 3xz^2 - y\} \) is irrotational or rotational by computing its curl. 2. The curl of a vecto

1. **State the problem:** Determine if the vector field $\mathbf{v} = \{6xy + z^3, 3x^3 - z, 3xz^2 - y\}$ is irrotational or rotational by computing its curl. 2. **Recall the formu