Given the vector field \(\vec{F} = (y \cos(xy) - 2x) \, \vec{i} + (x \cos(xy) + 2y) \, \vec{j}\):(a) Show that the vector field is a gradient vector field, \(\vec{F}\).(b) Find the potential function for the given vector field.(c) Evaluate the integral \(\int_L \vec{F} \cdot d\vec{r}\) where \(L\) is any curve connecting the points \(A = (\pi, 1)\) and \(B = (1, 2\pi)\).(d) What is the value of \(\oint_C \vec{F} \cdot d\vec{r}\) where \(C\) is a closed curve.

We fast so that F is a gradient vector field.

Answered: 5. Given the vector field F = (ycos(xy)…

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 r'(t), r(t), and r'(to) for the given value of to. r(t) = (et, e²t), to = 0 r'(t) = r(t) = r'(to) = Sketch the curve represented by the vector-valued function, and sketch the vectors r(to) and r'(to). y 3r O -1 eBook 2 (0, 1) r 1 r' 2 X 3 -1 لانا y 3 2 1 r (1, 0) 1 2 X 3 -1 y 2 1 r (1, 1) 1 2 X 3 -1 y 3 2 1 -1 (1, 1) 1 2 3 X
6. Denote by V the vector ·(). Then we can define the operations grad (f) = f (gradient of f), div (F) = V F (divergence of F), and curl (F) = ▼ × F (curl of F). (a) Given F = (2x + 3y, 3x + z, sin x), calculate div(F) and curl(F). (b) Is the vector field F = (x + y,3y, z-x+1) conservative? What about F = (yz, xz, xy)? (c) Calculate V × (Vf) =curl(grad f) and V. (V × F) = div(curl F).
In the images below, the vector field F is shown by the blue arrows and the curve Cis shown in red. Indicate the value of F ·dT. 1. /, F ·dÌ = 0 2. ,F ·dT > 0 3. F dT < 0 4. The value of F ·dT cannot be determined with only this information. X
3. (b) Let f(x, y) = log((ry - 2)²), u =(³, 4). i. Calculate the directional derivative Vaf (3, 1). ii. Determine the unit vector (direction) u for which Vaf (3, 1) is maximal.
8. A particle follows a straight-line path from (1,2) to (5,7) within the vector field F(x, y) = (xy, y²). Find the work. (That is, find f F dr where C is the path of the particle.)
Verify if the vector field F = (x² cos(3y), -³ sin(3y)) is conservative and evaluate the line integral Je where C' is the curve parameterized by R= for 0 < t < 1. cos(6) 3 A. I = B. I O c. I - OD. I = 0 E. I sin(6) 4 23 cos(3) 2 F. dR
Use the equation giving the flux of the vector field across the curve to calculate the flux of x+1 y (₁. (x + 1)² + y²' (x + 1)² + y² F(x, y): across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) [F C = Incorrect F. dr = 0.1074

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