Stokes' Theorem Equation - Explore our Question Collection

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

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

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.

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?

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)

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.

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

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 $x^{2} + y^{2} = 9$ .

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

Braxton Pugh 2021-01-04

Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes’ Theorem becomes the circulation form of Green’s Theorem.

Maiclubk 2021-01-02

Consider the vector field $F = ⟨ 5 z, x, 5 y ⟩$ and the surface which is the part of the elliptic paraboloid $z = x^{2} + 5 y^{2}$ that lies below the plane z = 5. Calculate curl(F) and then apply Stokes' Theorem to compute the exact magnitude of the flux of curl(F) through the surface using line integral. You do not need to cinfirm your answer by evaluating the double integral of curl(F) over the surface(the right-hand side of Stokes' Theorem).

generals336 2020-12-29

Use the Divergence Theorem to find the flux of $F = x y^{2} i + x^{2} y j + y k$ outward through the surface of the region enclosed by the cylinder $x^{2} + y^{2} = 1$ and the planes z = 1 and z =-1.

Ramsey 2020-12-17

Let $F (x, y) = ⟨ x y^{2} + 8 x, x^{2} y - 8 y ⟩$ . Compute the flux $\oint F \cdot n d s$ of F across a simple closed curve that is the boundary of the half-disk given by $x^{2} + y^{2} \leq 7, y \geq 0$ using the vector form Greens

OlmekinjP 2020-12-16

Let $f = [x^{2} y^{2}, - \frac{x}{y^{2}}]$ and $R : 1 \leq x^{2} + y^{2}, + 4, x \geq 0, y \geq x$ . Evaluate $\int_{C} F (r) \cdot d r$ counterclockwise around the boundary C of the region R by Green's theorem.

Reeves 2020-12-13

Use Green's Theorem to evaluate the line integral.
$(2 x + \ln 2) d y - (8 y^{2} + sinhx) d x$

necessaryh 2020-12-06

Use the Divergence Theorem to calculate the surface integral $\int \int_{S} F \cdot d S$ , that is, calculate the flux of F across S.
$F (x, y, z) = (\cos (z) + x y^{2}) i + x e^{- z} j + (\sin (y) + x^{2} z) k$
S is the surface of the solid bounded by the paraboloid $z = x^{2} + y^{2}$ and the plane z = 9.

foass77W 2020-12-03

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
$\int_{C} x y^{2} d x + 4 x^{2} y d y$
$C$ is the triangle with vertices $(0, 0), (3, 3)$ , and $(3, 6)$

pancha3 2020-12-02

Evaluate the line integral $\oint_{C} x y d x + x^{2} d y$ , where C is the path going counterclockwise around the boundary of the rectangle with corners (0,0),(2,0),(2,3), and (0,3). You can evaluate directly or use Greens

geduiwelh 2020-12-02

Green’s Theorem, flux form Consider the following regions R and vector fields F.
a. Compute the two-dimensional divergence of the vector field.
b. Evaluate both integrals in Green’s Theorem and check for consistency.
$F = ⟨ x, y ⟩, R = {(x, y) : x^{2} + y^{2} \leq 4}$

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