📘 multivariable calculus

1. **State the problem:** We want to evaluate the triple integral $$\iiint_V \phi \, dV$$ where $$\phi = 4xz$$ and $$V$$ is the volume bounded by the planes $$2z + 4y + x = 2$$ and

1. **State the problem:** Find the maximum and minimum values of the function $f(x,y,z) = xyz$ subject to the constraint $g(x,y,z) = xy + xz + yz = 108$. 2. **Method:** Use Lagrang

1. **State the problem:** We have a function $$Z = e^x \sin y$$ where $$x = u v^2$$ and $$y = v u^2$$.

1. **State the problem:** We want to evaluate the double integral $$\iint_D x \, dA$$ where $D$ is the region in the first quadrant between the circles $$x^2 + y^2 = 4$$ and $$x^2

1. **State the problem:** We want to evaluate the double integral $$\iint_R \arctan\left(\frac{y}{x}\right) dA$$ where the region $$R = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 4, 0 \leq

1. **Problem statement:** Given $V = r^m$ where $r^2 = x^2 + y^2 + z^2$, prove that $$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\p

1. **Problem Statement:** Given

1. **Find $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$ for $u = x^2 - y^2$, and evaluate at $(-2,-2)$.** - The function is $u = x^2 - y^2$.

1. **Problem 8:** Given $x = r \cos \theta$ and $y = r \sin \theta$, find $\frac{\partial r}{\partial x}$. 2. **Step 1:** Express $r$ in terms of $x$ and $y$. Since $x = r \cos \th

1. **Problem Statement:** Evaluate the triple integral $$\int_0^2 \int_0^{\sqrt{4-x^2}} \int_{-5+x^2+y^2}^{5-x^2-y^2} x \, dz \, dy \, dx$$ using polar coordinates, where the upper

1. **Problem Statement:** Evaluate the triple integral $$\int_0^2 \int_0^{\sqrt{4-x^2}} \int_{-5+x^2+y^2}^{3-x^2-y^2} x \, dz \, dy \, dx$$ by converting to polar coordinates, with

1. **Problem Statement:** We are given a density function $$\rho(x,y,z) = \frac{2xe^{xy}}{(z^3+2)z}$$ defined inside a cubic block with dimensions 2m by 2m by 2m, where $$x,y,z \in

1. **Problem Statement:** Evaluate the triple integral $$\iiint \frac{dx\,dy\,dz}{\sqrt{1 - x^2 - y^2 - z^2}}$$

1. **State the problem:** Find the directional derivative of the function $f(x,y) = x^2 y$ at the point $P = (4,6)$ in the direction of the vector $\vec{v} = 4\vec{i} - 3\vec{j}$.

1. You asked for help with multivariable calculus questions step by step. 2. Please provide the specific multivariable calculus problem you want to solve.

1. **Problem Statement:** Evaluate the line integral of the function $f(x,y,z) = x - 3y^2 + z$ over the line segment $C$ joining the points $O(0,0,0)$ to $A(1,1,1)$.

1. **Problem statement:** Given a function $F$ of variables $x$ and $y$, where $x = e^u \sin v$ and $y = e^u \cos v$, prove that $$\frac{\partial^2 F}{\partial x^2} + \frac{\partia

1. **State the problem:** Find the Jacobian determinant $\frac{\partial(u,v,w)}{\partial(x,y,z)}$ where $$u = x^2 - 2y, \quad v = x + y + z, \quad w = x - 2y + 3.$$

1. **Problem statement:** Given the function $$u = \frac{x^2 + y^2}{x + y},$$ prove that $$\left(\frac{\partial u}{\partial x} - \frac{\partial u}{\partial y}\right)^2 = 4 \left(1

1. **Problem statement:** Evaluate the double integral $$\iint_R \tan^{-1}\left(\frac{y}{x}\right) \, dA$$ where $R$ is the region inside the circle $$x^2 + (y-1)^2 = 1$$ and outsi

1. **State the problem:** We want to find the limit of the function $$f(x,y) = \frac{xy}{\sqrt{x^2 + y^2}}$$ as the point $$(x,y)$$ approaches $$(0,0)$$. 2. **Recall the definition