📘 vector calculus

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

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 = \