📘 vector calculus

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,

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