Multivariable Calculus Examples: Learn Complex Calculus Problems

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Calculus and Analysis
Multivariable calculus

Recent questions in Multivariable calculus

Jason Farmer 2021-01-31

Nonexistence of a limit Investiage the limit $lim_{(x, y) \to (0, 0)} \frac{{(x + y)}^{2}}{x^{2} + y^{2}}$

generals336 2021-01-31

What tis the complete domain D and range R of the following multivariable functions:
$w = 2 \sin x y$

coexpennan 2021-01-31

Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use.
$F = ⟨ x \sin y, - \cos y, z \sin y ⟩$ , S is the boundary of the region bounded by the planes x = 1, y = 0, $y = \frac{π}{2}, z = 0$ , and z = x.

Carol Gates 2021-01-31

use Green’s Theorem to find the counterclockwise circulation and outward flux for the field F and the curve C. $F = (x - y) i + (y - x) j$
C: The square bounded by $x = 0, x = 1, y = 0, a n d y = 1$

texelaare 2021-01-27

Write formulas for the indicated partial derivatives for the multivariable function.
$g (k, m) = k^{4} m^{5} - 3 k m$
a) $g_{k}$
b) $g_{m}$
c) $g_{m} ∣_{k = 2}$

FizeauV 2021-01-27

Proof of Stokes’ Theorem Confirm the following step in the proof of Stokes’ Theorem. If z = s(x, y) and f, g, and h are functions of x, y, and z, with M = f + hz_x and $N = g + h z_{y}$ , then
$M_{y} = ƒ_{y} + ƒ_{z} z_{y} + h z_{x y} + z_{x} (h y + h_{z} z_{y})$
and
$N_{x} = g_{x} + g_{z} z_{x} + h z_{y x} + z_{y} (h_{x} + h_{z} z_{x})$ .

usagirl007A 2021-01-19

Using the Divergence Theorem, evaluate $\int \int_{S} F . N d S$ , where $F (x, y, z) = (z^{3} i - x^{3} j + y^{3} k)$ and S is the sphere $x^{2} + y^{2} + z^{2} = a^{2}$ , with outward unit normal vector N.

Carol Gates 2021-01-16

True or false?
Given an oriented piecewise-smooth surface that is bounded by a simple, closed, piecewise-smooth boundary curve with positive orientation, and a vector field whose components have continuous partial derivatives on an open region in three-dimensional Euclidean space. Then the line integral of this vector field along the boundary curve equals the flux of the curl of the vector field across the surface.

babeeb0oL 2021-01-15

Using the two approaches below, evaluate the line integral. y) dx + (x+y)dy C os counerclockwise around the circle with center the origin and radius 3(a) directly (b) using Green's Theorem.

Mylo O'Moore 2021-01-13

Use Green's Theorem to evaluate $F \cdot d r$ . (Check the orientation of the curve before applying the theorem.)
$F (x, y) = \sqrt{x} + 4 y^{3}, 4 x^{2} + \sqrt{y}$
C consists of the arc of the curve $y = \sin (x)$ from (0, 0) to $(π, 0)$ and the line segment from $(π, 0)$ to (0, 0)

Falak Kinney 2021-01-13

Average value over a multivariable function using triple integrals. Find the average value of $F (x, y, z) = x^{2} + y^{2} + z^{2}$ over the cube in the first octant bounded bt the coordinate planes and the planes $x = 5$ , $y = 5$ , and $z = 5$

Anonym 2021-01-13

Stokes

permaneceerc 2021-01-13

How do Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem relate to the Fundamental Theorem of Calculus for ordi-nary single integral?

Amari Flowers 2021-01-10

Let $F \vec{\to} r = < x, y, z >$ and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).

Tobias Ali 2021-01-07

Let $v = ⟨ 4 - x \cos z - x e^{z} \cos y, y \cos z, e^{z} \cos y ⟩$ be the velocity field of a fluid. Compute the flux of v across the surface $x^{2} + y^{2} + z^{2} = 9$ where x > 0 and the surface is oriented away from the origin.

EunoR 2021-01-06

Use Stokes' Theorem to evaluate $\int_{C} F \cdot d r$ where C is oriented counterclockwise as viewed above.
$F (x, y, z) = x y i + 3 z j + 5 y k$ , C is the curve of intersection of the plane $x + z = 10$ and the cylinder $x^{2} + y^{2} = 9$ .

Ava-May Nelson 2021-01-06

Let $F (x, y) = ⟨ 5 \cos (y), 8 \sin (y) ⟩$ .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

Anish Buchanan 2021-01-06

Let $z (x, y) = e^{3 x y}, x (p, q) = \frac{p}{q} and y (p, q) = \frac{q}{p}$ are functions. Use multivariable chain rule of partial derivatives to find
(i) $\frac{\partial z}{\partial p}$
(ii) $\frac{\partial z}{\partial q}$ .

ddaeeric 2021-01-05

Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3).
Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle $x^{2} + y^{2} = 1$ from(0,1)to(1,0).

babeeb0oL 2021-01-05

Let F = curl(A). Which of the following are related by Stokes' Theorem? (a) The circulation of A and flux of F (b) The circulation of F and flux of A

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!

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