2. Given that F(x, y, z) = (e", xe + e*, ye²) is a conservative vector field.a. Find a potential function f for F.b. Compute the line integral foF dr where C is the arc of the circular helix givenbyr(t) = cos ti+ sin tj+ tk, 0≤t≤ π/2.

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Answered: 2. Given that F(x, y, z) = (e", xe" +…

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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B2: Assume K is a body (Area) that is coherent. Let C be the curve that is parametised by the following equation (tan () .cos"(), sin°(1) ) osisa. Assume the following vector field thet is well defined in "K" and "a" is the value that gives Fa potential V(x, y, z). axz 1 + 2r, x²z+ y + 2y, + 2z 1²z + y° F(x, y, z) = 1²z + y_ a) Calculate a) b) Calculate the lineintegral F - dr.
Find the line integral of the vector field F= over the curve C which is the quarter of the unit circle from P(0,1) to Q(1,0).
Explain in your own words what does “divergence of vector field” mean. Calculate V. f, where f = sin(x) a + sin(y) ay. (The vectors a, ay are unit vec- tors.). Plot the divergence function. What do the maximum and minimum values represent?
Along the curve C traced out by the end point of the position vector tk (t) = (- sin ti + cos tj+t) = [0,π/2] calculate the line integral √ F(r) - di dr of the vector field F(x, y, z) = −y+xj+yzk.
3. Let D be the upper half-plane (y > 0) and consider the vector field F: D+R² given by F(x, y) = (7, =7). Test whether F is a gradient field, and if so, find a potential P.
Compute the divergence and curl of the vector field cos? (x2 ) F = sin (²y=) 2. 10 yz4 – ryz , at the point (1, 1, c). a. Divergence = ab sin (a) f Ω b. Curl = sin (a) f Ω a 00
5. Given the vector field F(x, y, z) = (e», (xe" + e²), ye²) find the following a. Show F is conservative b. Find a potential function f (i.e. F = Vf) c. Find the line integral S. F · dř where C is the line segment from (0,2,0) to (4,0,3)
3. Determine if the vector field F(x, y, z) = e" sin yzi + ze² cos yzj + ye² cos yzk is conservative or not. If it is, find the potential function f such that F = Vf.

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2. Given that F(x, y, z) = (e", xe" + e*, ye*) is a conservative vector field. a. Find a potential function f for F. b. Compute the line integral Jo F. dr where C is the arc of the circular helix given by r(t) = cos ti+ sin tj + tk, 0≤t≤T/2.

2. Given that F(x, y, z) = (e", xe" + e, ye) is a conservative vector field. a. Find a potential function f for F. b. Compute the line integral Jo F. dr where C is the arc of the circular helix given by r(t) = cos ti+ sin tj + tk, 0≤t≤T/2.