Stokes' Theorem Equation - Explore our Question Collection

College
Calculus and Analysis
Multivariable calculus
Green's, Stokes', and the divergence theorem

Recent questions in Green's, Stokes', and the divergence theorem

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 $y = x^{2} and y = 4$ .

Cabiolab 2021-02-25

Evaluate $\int_{C} x^{2} y^{2} d x + 4 x y^{3} d y$ where C is the triangle with vertices(0,0),(1,3), and (0,3).
(a)Use the Greens

sagnuhh 2021-02-25

Use Stokes' Theorem to evaluate $\int_{C} F \cdot d r$ where C is oriented counterclockwise as viewed from above.
$F (x, y, z) = (x + y^{2}) i + (y + z^{2}) j + (z + x^{2}) k$ ,
C is the triangle with vertices (3,0,0),(0,3,0), and (0,0,3).

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

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 $x^{2} + 4 y^{2} = 4$

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

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 $x = 0, x = 3, y = 0, y = 3, a n d z = 0, z = 3$ .

Ayaana Buck 2021-02-12

Explain carefully why Greens

facas9 2021-02-09

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

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.

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 $W = 12.56 .$

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.

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 $x = 0, x = 1, y = 0, a n d y = 1$

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.

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.

If you have studied calculus and analysis extensively, the chances are high that you have discovered problems on green's theorem that leaves more questions than answers for an average person. Keeping this fact in mind, we have provided Stokes theorem example that will help you find the most efficient solutions as you look through the list of stokes theorem questions based on equations and mathematical analysis. At the same time, you can compare various applications of Stoke's theorem problem like in control of the magnetic fields or the water flow.

Multivariable calculus

Master Green's Theorem Problems with Plainmath