📘 multivariable calculus

1. **Stating the problem:** We are given a system of three equations in four variables $(x,y,z,t)$:

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

1. **State the problem:** Find the gradient of the function $f(x,y) = \cos(x^2 + y^2)$ at the point $(3, -4)$. 2. **Recall the gradient formula:** The gradient of a function $f(x,y

1. **Problem Statement:** Find the volume of the solid region $D$ bounded below by the cone $z=\sqrt{x^2+y^2}$ and above by the plane $z=1$ using spherical coordinates.

1. **Problem statement:** Sketch the region of integration for the integral $$\int_0^1 \int_0^{\sqrt{1+x^2}} f(x,y) \, dy \, dx$$

1. **Problem Statement:** We are given a solid region $Q$ bounded by:

1. **State the problem:** We want to find the limit of the function $$f(x,y) = \begin{cases} \frac{x^3 - y^3}{x + y} & \text{if } x + y \neq 0 \\ 0 & \text{if } x + y = 0 \end{case

1. **State the problem:** Find the volume of the solid tetrahedron enclosed by the plane $2x + y + z = 4$ and the coordinate planes $x=0$, $y=0$, and $z=0$ using a triple integral.

1. **Problem Statement:** We are given two spheres and bounding planes:

1. **Problem statement:** We need to evaluate the triple integral $$\iiint_Q \frac{\sqrt[3]{x^2 + y^2 + z^2}}{2} \, dV$$ using spherical coordinates. 2. **Recall spherical coordina

1. The problem is to understand the concept of limits in multivariable calculus. 2. The limit of a function $f(x,y)$ as $(x,y)$ approaches a point $(a,b)$ is defined as $$\lim_{(x,

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: