📘 multivariable calculus

1. The first problem is to find the partial derivatives $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$ for the function $f(x,y) = \frac{y}{x^2 + y^2}$ whe

1. **Problem statement:** Find the points where the function $f(x,y) = x + y$ attains its maximum and minimum values subject to the constraint $x^2 + y^2 = 1$. 2. **Method:** Use t

1. **Stating the problem:** We want to find the result of the second iteration using Newton's method for the function

1. **Problem Statement:** Calculate the double integral $$\iint_R f(x,y) \, dA$$ where $$R = \{(x,y): 0 \leq x \leq 4, 0 \leq y \leq 2\}$$ and the function $$f(x,y)$$ is defined as

1. Given $u = f(x - y, y - z, z - x)$, prove that $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = 0.$$ 2. Use chain ru

1. The problem is to find the third mixed partial derivative $f_{rst}$ of the function $f(r,s,t) = e^r \sin(st)$. 2. First, recall the function: $$f(r,s,t) = e^r \sin(st)$$

1. **Problem:** Compute the Jacobian \( \frac{\partial(u,v)}{\partial(x,y)} \) where \( u = x + y \) and \( v = xy + 1 \). 2. **Formula:** The Jacobian determinant for functions \(

1. The problem asks to find a unit vector in the direction where the function $f(x,y) = 6x^6 y^7$ increases most rapidly at the point $P(-1,1)$, and to find the rate of change of $

1. **State the problem:** We need to find the directional derivative of the temperature function $$T = x^3 y + y^3 z + z^3 x$$ at the point $P = (2, -1, 0)$ in the direction toward

1. The problem is to understand and compute surface and volume integrals. 2. Surface integrals calculate the integral of a function over a surface $S$. The formula is $$\iint_S f(x

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