📘 vector calculus

1. **Problem statement:** Verify Stokes' theorem for the vector field $\vec{A} = (y+z)\hat{i} - xz\hat{j} + y^2\hat{k}$ over the surface $S$ in the first octant bounded by $2x + z

1. **State the problem:** Verify Stokes' theorem for the vector field $\vec{A} = (y+z)\hat{i} - xz\hat{j} + y^2\hat{k}$ over the surface $S$ in the first octant bounded by $2x + z

1. **Problem:** Calculate the line integral \(\int_c \vec{F} \cdot d\vec{r}\) where \(\vec{F} = (x^2 + y) \vec{i} + (y^2 - x) \vec{j}\) and \(c\) is the quarter circle \(x^2 + y^2

1. **Problem statement:** Find the surface integral \(\int_{\Sigma} \vec{F} \cdot d\vec{S}\) where \(\Sigma\) is the hemisphere of radius 1 centered at the origin, lying above the

1. **Problem statement:** Find the work done by the force field \(\vec{F} = (x^3 - 3xy^2) \vec{i} + (3x^2 y - y^3) \vec{j}\) moving an object from \(A(1,0)\) to \(B(0,1)\) along th

1. The problem asks to sketch the vector field for the function \( \mathbf{F}(x,y) = 2\mathbf{i} - \mathbf{j} \).\n\n2. This vector field assigns the vector \( 2\mathbf{i} - \mathb

1. **Problem:** Prove that $$\frac{d}{dt} e_{\rho} = \dot{\phi} e_{\phi}, \quad \frac{d}{dt} e_{\phi} = -\dot{\phi} e_{\rho}$$ where dots denote differentiation with respect to tim

1. **State the problem:** We want to evaluate the line integral \( \oint_{\partial D} \mathbf{F} \cdot d\mathbf{r} \) where \( \mathbf{F}(x,y) = (P(x,y), Q(x,y)) \) with $$

1. **Problem:** Find the cross products $\mathbf{a} \times \mathbf{b}$ and $\mathbf{b} \times \mathbf{a}$ for $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, 5, 6)$.\n\n2. **Formula

1. **State the problem:** Find the unit tangent vector $\mathbf{T}$, the principal unit normal vector $\mathbf{N}$, and the unit binormal vector $\mathbf{B}$ for the curve given by

1. **State the problem:** We need to find the unit tangent vector $\mathbf{T}$, the principal unit normal vector $\mathbf{N}$, and the unit binormal vector $\mathbf{B}$ for the cur

1. **Problem Statement:** Given the curve $$\mathbf{r}(t) = \frac{2}{3}(1 - t)^{3/2} \mathbf{i} + t \mathbf{j} + \frac{2}{3} t^{3/2} \mathbf{k}, \quad 0 < t < 1,$$

1. **Problem:** Find the unit vector normal to the surface $x^2 y + 2 x z^2 = 8$ at the point $(1,0,2)$. 2. **Formula and rules:** The normal vector to a surface defined by $F(x,y,

1. Masalani bayon qilamiz: $X=\{y+z, x+z, x+y\}$ vektor maydon konform vektor maydonmi? 2. Konform vektor maydon uchun, vektor maydonning gradienti simmetrik va uning rotatsiyasi n

1. **State the problem:** Evaluate the line integral $$\int_C \mathbf{F} \cdot d\mathbf{r}$$ where $$\mathbf{F} = \cos y \hat{i} + x \sin y \hat{j}$$ and $$C$$ is the curve $$y = \

1. **Problem Statement:** Given the space curve defined by $$x = \cos t, \quad y = \sin t, \quad z = 3t,$$ show that the curvature $$\kappa = \frac{1}{10}$$ and the torsion $$\tau

1. The problem is to understand and sketch the vector field given by $$\mathbf{F}(x,y) = \left\langle -\frac{y}{2}, \frac{x}{2} \right\rangle.$$\n\n2. This vector field assigns to

1. **Problem statement:** Given the vector equation $\mathbf{v} = \mathbf{w} \times \mathbf{r}$, where $\mathbf{w}$ is a constant vector and $\mathbf{r}$ is the position vector, pr

1. **Problem statement:** Given the vector equation $\mathbf{v} = \mathbf{w} \times \mathbf{r}$, where $\mathbf{w}$ is a constant vector and $\mathbf{r}$ is the position vector, pr

1. **State the problem:** We need to find the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ where $\mathbf{F} = x^2 \mathbf{i} - yz \mathbf{j} + x \cos z \mathbf{k}$ and the

1. **Problem Statement:** Verify Green's theorem for the given line integrals and curves, and explain its application in mesh processing.