Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C.F= (4x + ex siny)i + (5x + ex cosy) jC: The right-hand loop of the lemniscate r² = cos 20Describe the given region using polar coordinates. Choose 0-values between - and .≤0≤≤rs √cos (20)(Type exact answers.)The outward flux is(Type an integer or a simplified fraction.)The counterclockwise circulation is.(Type an integer or a simplified fraction.)

Answered: Image /qna-images/answer/9073e5b0-2c45-4c9e-96bc-62e12bb37774.jpg

Answered: Use Green's Theorem to find the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the gradient and Laplacian of the scalar field f in two dimensions, given in the Cartesian coordinates as: f(x, y) = y sin(x), and show that the Laplacian of f is equal to f. Find the directional derivative of f at the point P(π/6, 5) in the direction of a = 3i+4j.
Consider the vector field F and the curve C below. F(x, y) (3 + 8xy2)i + 8x²yj, C is the arc of the hyperbola y = 1/x from (1, 1) to (4, = (a) Find a potential function f such that F = Vf. f(x, y) = (b) Use part (a) to evaluate Sc Vf. dr along the given curve C.
Consider the field F = (x + y)i - (x² + y2), and the counterclockwise curve C whose boundary is defined by the lines y = 0, x = 2, and y = x. (a) Sketch a graph of the curve C described above. Indicate the direction on the graph. (b) Express the circulation of the field F as a double integral using Green's theorem over C. Evaluate the integral to find the circulation. (c) Express the flux of the field F as a double integral using Green's theorem over C. Evaluate the integral to find the flux.
Find the gradient and Laplacian of the scalar field f in two dimensions, given in the Cartesian coordinates as: f(x, y) = y sin(x), and show that the Laplacian of f is equal to f. Find the directional derivative of f at the point P(π/6,5) in the direction of a = 3i+4j.
Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C. F = (xy+3y²)i + (x-3y)j CEER The outward flux is (Type an integer or a simplified fraction.) Get more help - 67°F Light rain Esc Tab Caps Lock Shift 1 ~ ↑ F1 Q A O 2 N @ W F3 O + 3 S # X F4 Q E 4 FS $ Clear all = - H R C 5 (0,0) % LL F x=y² T y=x² 6 (1,1) Check answer FB CA > Q Q 7 Y & H B ¥ 0012 8
Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C. F=9x³y²i+x²yj www The outward flux is (Type an integer or a simplified fraction.) Get more help - 67°F 3 Light rain Esc Tab Caps Lock Shift ↑ 1 F. - I A F2 ☀ - @ 2 N W F3 S Q+ 3 # F4 ¹0 E 4 $ FS R Clear all - H D x C (0,0) 5 LL % y=x F Check answer OFDH C F6 F7 FB C y=x²-2x T 6 Ga > (3,3) G Q Q 7 Y & LG F9 H в * 8
(a) Set up a double integral for calculating the flux of the vector field F(x, y, z) = xi+yj through the open-ended circular cylinder of radius 4 and height 7 with its base on the xy-plane and centered about the positive z-axis, oriented away from the z-axis. If necessary, enter as theta. Flux = A = B = - IT² C = D= (b) Evaluate the integral. Flux = [fF.d F.dA= dz de

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Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C. = (4x + ex sin y)i + (5x + ex cos y) j C: The right-hand loop of the lemniscate r² = cos 20