Multivariable Calculus Examples: Learn Complex Calculus Problems

College
Calculus and Analysis
Multivariable calculus

Recent questions in Multivariable calculus

CMIIh 2021-02-25

Use Green's Theorem to evaluate $\int_{C} (e^{x} + y^{2}) d x + (e^{y} + x^{2}) d y$ where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by $y = x^{2} and y = 4$ .

vazelinahS 2021-02-25

Let $F (x, y) = ⟨ 4 \cos (y), 2 \sin (y) ⟩$ . Compute the flux $\oint F \cdot n d s$ of F across the boundary of the rectangle $0 \leq x \leq 5, 0 \leq y \leq \frac{π}{2}$ using the vector form of Greens

naivlingr 2021-02-22

Write formulas for the indicated partial derivatives for the multivariable function. $g (x, y, z) = 3.1 x^{2} y z^{2} + 2.7 x^{y} + z$
a) $g_{x}$
b) $g_{y}$
c) $g_{z}$

Elleanor Mckenzie 2021-02-22

Multivariable optimization. Find the demisions of the rectangular box with largest volume if the total surface are is given as 64 $c m^{2}$

Sinead Mcgee 2021-02-21

Let $F = [x^{2}, 0, z^{2}]$ , and S the surface of the box $| x | \leq 1, | y | \leq 3, 0 \leq z \leq 2$ .
Evaluate the surface integral $\int \int_{S} F \cdot n d A$ by the divergence theorem.

nicekikah 2021-02-21

If E(t,x,y,z) and B(t,x,y,z)represent the electric and magnetic fields at point (x,y,z) at time t, a basic principle of electromagnetic theory says that $\nabla \times E = \frac{- \partial B}{\partial t}$ . In this expression $\nabla \times E$ is computed with t held fixed and $\frac{\partial B}{\partial t}$ is calculated with (x,y,z) fixed.
Use Stokes

aortiH 2021-02-21

Use Green's Theorem to evaluate
$\oint_{C} (x^{2} + y) d x - (3 x + y^{3}) d y$
Where $C$ is the ellipse $x^{2} + 4 y^{2} = 4$

alesterp 2021-02-20

Set up the integral for the divergence theorem both ways. Then find the flux.
$F (x, y, z) = 3 x \hat{i} + x y \hat{j} + 2 x z \hat{k}$
E is the cube bounded by the planes $x = 0, x = 3, y = 0, y = 3, a n d z = 0, z = 3$ .

Albarellak 2021-02-19

Find the critical points of the following multivariable function, and using the Second Derivative Test to find the maxima and minima of the equation below: $a (x, y) = \frac{8 x - y}{e^{x^{2} + y^{2}}}$

boitshupoO 2021-02-19

Multivariable optimization question. Find three positive real numbers whose sum is one and the sum of their squares is a minimum.

ruigE 2021-02-18

A surface is represented by the following multivariable function,
$f (x, y) = x^{3} + y^{3} - 3 x - 3 y + 1$
Calculate coordinates of stationary points.

Ayaana Buck 2021-02-12

Explain carefully why Greens

glasskerfu 2021-02-09

A particle moves along line segments from the origin to the points(3,0,0),(3,3,1),(0,3,1), and back to the origin under the influence of the force field $F (x, y, z) = z^{2} i + 3 x y j + 4 y^{2} k$ .
Use Stokes' Theorem to find the work done.

facas9 2021-02-09

What is the Divergence Theorem? Explain how it generalizes Green’s Theorem to three dimensions.

snowlovelydayM 2021-02-09

Consider this multivariable function. $f (x, y) = x y + 2 x + y - 36$
a) What is the value of $f (2, - 3)$ ?
b) Find all x-values such that $f (x, x) = 0$

Isa Trevino 2021-02-05

Use Stokes' Theorem to compute $\oint_{C} \frac{1}{2} z^{2} d x + (x y) d y + 2020 d z$ , where C is the triangle with vertices at(1,0,0),(0,2,0), and (0,0,2) traversed in the order.

shadsiei 2021-02-05

Let C be the ellipse contained in the xy plane whose equation is $4 x^{2} + y^{2} = 4$ , oriented clockwise. The force field F described by $F (x, y, z) = x^{2} i + 2 x j + z^{2} k$ , moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: $z = 4 - 4 x^{2} - y^{2}, z \geq 0$ (with ascending orientation, that is, the component in the z direction equal to 1) and assuming $π = 3.14$ , we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore $W = 0$ .
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that $W = 12.56 .$
c) We can apply Stokes' Theorem and conclude that $W = 6.28$
d) We can apply Stokes' Theorem and conclude that $W = 12.56 .$

alesterp 2021-02-05

Use polar coordinates to find the limit. [Hint: Let $x = r \cos and y = r \sin$ , and note that (x, y) (0, 0) implies r 0.] $lim_{(x, y) \to (0, 0)} \frac{x^{2} - y^{2}}{\sqrt{x^{2} + y^{2}}}$

Khadija Wells 2021-01-31

Find the region that one of the cycloid's arcs borders $x = a (0 - \sin (0)), y = a (1 - \cos (0))$ ,where a > 0, and $0 \leq 0 \leq 2 π$ , and the axis(use Green's theorem).

Anish Buchanan 2021-01-31

Calculate the surface integral using the divergence theorem. $\int \int_{S} F \cdot d S$ , i.e., determine the flux of F across S.
$F (x, y, z) = (x^{3} + y^{3}) i + (y^{3} + z^{3}) j + (z^{3} + x^{3}) k$ , S is a sphere with an origin-centered center and a radius of 2.

As you start exploring calculus and analysis, you will encounter multivariable calculus equations that are self-explanatory as well because all of them will contain at least two questions related to each variable being involved. See our multivariable calculus examples to receive more help and information regarding how these are used. The answers to solving these multivariable calculus questions should be based on finding the deterministic behavior. These are used in engineering and those fields where the parametric equations solver will provide you an optimal control of time dynamic systems. As an interesting subject, applying at least one equation in practice will keep you inspired!

Multivariable calculus

Become a Multivariable Calculus Pro!