📘 multivariable calculus

1. The problem asks to evaluate the double integral over the entire plane of the vector function \(\begin{pmatrix}0 \\ y \\ x\end{pmatrix}\) with respect to \(x\) and \(y\) from \(

1. **Problem statement:** Given the function $$f(x,y) = \frac{1}{3}x^3 + \frac{4}{3} y^3 - x^2 - 3x - 4y - 2025,$$ we need to find partial derivatives, critical points, the discrim

1. **State the problem:** We want to find the Jacobian of a transformation scaled by the determinant given a vector and then evaluate an integral involving the variables.

1. **Problem statement:** Given the function $$U = \log(x^3 + y^3 + z^3 - 3xyz),$$ prove that

1. **State the problem:** Find the extrema values of the function $$f(x,y,z) = x^2 + 2y^2 + 3z^2$$ subject to the constraints $$x + y + z = 1$$ and $$x - y = 0$$. 2. **Use the meth

1. **State the problem:** Find the critical points of the function $$f(x,y) = x^2 y^2 - x^2 - y^2$$ and classify each as a relative minimum, relative maximum, or saddle point. 2. *

1. **Problem:** Check the continuity of the function $$f_1(x,y) = \begin{cases} \frac{x^2 + y^2}{\sqrt{x^2 + y^2 + 1} - 1} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$

1. **State the problem:** Evaluate the double integral $$\int_0^1 \int_1^3 (1 + 8xy) \, dy \, dx.$$\n\n2. **Recall the formula for double integrals:** The integral over a rectangul

1. **Problem:** Find the volume of the solid bounded by $z=0$ and $z=4-r^2$ where $r=\sqrt{x^2+y^2}$, using cylindrical coordinates, given the circle $x^2 + y^2 = 9$. 2. **Formula

1. **Problem 1:** Find the Jacobian $\frac{\partial(x,y)}{\partial(u,v)}$ for $x = -5u - 3v$, $y = -4u - 3v$. 2. The Jacobian determinant for two variables is given by:

1. **Problem:** Consider the surface in $\mathbb{R}^3$ parameterized by $\vec{\Phi}(r, \theta) = (r \cos \theta, r \sin \theta, \theta)$ with $0 \leq r \leq 1$ and $0 \leq \theta \

1. The problem is to describe a helicoid-like spiral surface extending from $z=0$ to $z=4\pi$ with radius $r$ from 0 to 1, wrapping twice around the $z$-axis. 2. A helicoid can be

1. **Problem:** Consider the surface in $\mathbb{R}^3$ parameterized by $\vec{\Phi}(r,\theta) = (r \cos \theta, r \sin \theta, \theta)$ with $0 \leq r \leq 1$ and $0 \leq \theta \l

1. **Problem statement:** Find the volume of the solid bounded by the cone $$z = \sqrt{x^2 + y^2}$$ and the paraboloid $$z = 6 - x^2 - y^2$$ using cylindrical coordinates. 2. **Coo

1. Problem: Find all local maxima and minima of $$f(x,y) = x^2 + 4y^2 - 2x + 8y - 1$$. Step 1: Find partial derivatives:

1. **Problem Statement:** Find all first and second partial derivatives of the function $$f(x,y) = \frac{xy}{x^2 + y^2}$$ and verify the mixed partial derivatives. 2. **Recall the

1. **Problem Statement:** Evaluate the integral of $f(x,y) = x^2 + y^2$ over the triangular region with vertices $(0,0)$, $(1,0)$, and $(0,1)$. 2. **Formula and Setup:** The integr

1. **Problem statement:** Find the volume under the surface $$z = 6 - x - y$$ above the triangular region bounded by $$x=0$$, $$y=0$$, and $$x + y = 2$$. 2. **Understanding the reg

1. **Problem 1:** Find the minimum and maximum values of $f(x,y) = xy$ subject to the constraint $2x^2 + 8y^2 = 16$. 2. **Method:** Use Lagrange multipliers. Set up the system:

1. **State the problem:** Given the function $$u=\frac{x^2 y^2 z^2}{x^2 + y^2 + z^2} + \cos\left(\frac{xy + yz}{x^2 + y^2 + z^2}\right),$$ show that $$x \frac{\partial u}{\partial

1. **Problem 1:** Interpret the limit \(\lim_{(x,y)\to(0,0)} \frac{x^2 - y^2}{x^3 + y^3}\) by considering limits along coordinate axes. - Along \(y=0\):