📘 multivariable calculus

1. **Problem:** Given $z = \tan(y + ax) + (y - ax)^{3/2}$, find $\frac{\partial^2 z}{\partial x^2} - a^2 \frac{\partial^2 z}{\partial y^2}$. 2. **Formula and rules:** Use partial d

1. **Problem statement:** Find the angle between the surfaces given by $$xy^2z = 3x + z^2$$

1. **State the problem:** Find the angle between the surfaces defined by $$xy^2z = 3x + z^2$$

1. **State the problem:** Verify if the differential equation $$z \, dx + z \, dy + (x + y + \sin z) \, dz = 0$$ is exact and if so, find its primitive function. 2. **Recall the co

1. **Problem Statement:** Find the volume under the plane $$3x + 2y - z = 0$$ and above the region enclosed by the parabolas $$y = x^2$$ and $$x = y^2$$. 2. **Understanding the reg

1. **Problem statement:** Calculate the line integral of the function $f(x,y,z) = x + \sqrt{y - z^2}$ along two given curves from $(0,0,0)$ to $(1,1,1)$. 2. **Recall the line integ

1. The problem is to understand the function $f(x,y,z) = x + y + z$ defined over a rectangular box in 3D space with corners at $(0,0,0)$ and $(1,2,3)$. 2. This function simply adds

1. **Problem statement:** We need to find the line integral of the scalar function $$f(x,y,z) = \frac{\sqrt{3}}{x^2 + y^2 + z^2}$$ along the curve defined by $$\mathbf{r}(t) = t\ma

1. **State the problem:** Find the absolute extrema of the function $$f(x,y,z) = 2x - 3y + z - 1$$ subject to the constraint $$x^2 + y^2 + z^2 = 14$$. 2. **Method:** Use Lagrange m

1. **Problem Statement:** Calculate the double integral $$\iint \frac{dx\,dy}{xu}$$ over the region bounded by the four circles given by the equations:

1. **State the problem:** Find the maximum and minimum values of the function $$f(x,y) = x^2 + y$$ subject to the constraint $$g(x,y) = x^2 + y^3 = 1$$ using Lagrange multipliers.

1. The problem asks for the geometric interpretation of the partial derivative $f_x(a,b)$ of a function $f(x,y)$ at the point $(a,b)$. 2. Recall that the partial derivative $f_x(a,

1. We are asked to find the point on the surface $z = xy + 1$ that is nearest to the origin $(0,0,0)$. 2. The distance $D$ from any point $(x,y,z)$ to the origin is given by the fo

1. **Problem Statement:** Correct and solve the first problem: Find the domain, global maximum, and global minimum of the function $$f(x,y) = \sqrt{64 - x^2 - y^2}$$.

1. **Problem statement:** Calculate the second partial derivatives $\frac{\partial^2 f}{\partial x^2}$, $\frac{\partial^2 f}{\partial \theta^2}$, and $\frac{\partial^2 f}{\partial

1. Diberikan fungsi dua peubah dengan persamaan $$z = f(x,y) := \sqrt{x^2 + y^2}$$. Kita diminta untuk menggambar grafik fungsi ini. 2. Fungsi ini merupakan fungsi jarak dari titik

1. **Problem Statement:** Given the function $$u = \frac{1}{\sqrt{x^2 + y^2}}$$, find the second partial derivatives $$\frac{\partial^2 u}{\partial x^2}$$ and $$\frac{\partial^2 u}

1. **Problem statement:** Find the volume of the solid in the first octant bounded by the coordinate planes ($x=0$, $y=0$, $z=0$), the cylinder $x^2 + y^2 = 4$, and the plane $z +

1. The problem asks to find the partial derivatives $f_u$, $f_v$, and $f_w$ of the function $$f(u,v,w) = e^{uv} \ln w.$$ 2. Recall the rules for partial derivatives: when different

1. **Problem Statement:** Given a function $u = f(x^2 + 2yz, y^2 + 2xz)$, find the value of $$z \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial z} - x \frac{\partial u

1. **State the problem:** Find all local maxima, minima, and saddle points of the function $$f(x,y) = \sqrt{56x^2 - 8y^2 - 16x - 31} + 1 - 8x$$. 2. **Domain consideration:** The ex