5) (a) Show thatF = (8-14xy² + 2ye2x) + (e2x - 14x²y) ŷis conservative.(b) Find a potential function for the field F defined in (a) usingthe line integral method.(c) Use the "partial integration" method to find a potentialfunction for F, checking that both methods agree.

Answered: Image /qna-images/answer/5980d90a-8007-4036-a874-19fa61acb315.jpg

Answered: 5) (a) Show that F = (8-14xy2 + 2ye²x)…

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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3) a) Use the Taylor theorem for function U(x, t) with the step size(−Ax ) (t, is held cons. b) Consider the second-order truncation error (0(Ax )²) with the step size (-Ax ) and then, obtain a finite difference approximation for the first-order derivative of U(x, t) respect x (Ux(x, t)) . c) Consider the third-order truncation error (0 (Ax )³) with the step size Ax and also (-Ax) then, subtract them and finally obtain a finite difference approximation for the first-order derivative of U respect x (Ux(x, t)). d) Consider the fourth-order truncation error (0(Ax )4) with the step size Ax and also (-Ax) then, add them and finally obtain a finite difference approximation for the second-order derivative of U respect x (Uxx (x, t)).
How do you complete this question?
do 2 b)d)j)
Compute the gradients of each of the following functions: o = xyz & = ln [(x² - y²)-¹/2] r = || = √√√x² + y² + x² 1/r Note: Remember that your tutorial has Hints in it, at the end of each chapter. If you're stuck, they are often quite useful! • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, shift and - for subscripts, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx= x², Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {A₂, Ay, A₂}. Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. Vo= V = Vr = V= (2A, 2A, 2A₂).
Let f : R²→ R be defined by: 1 f (x) (2x1 + x2 + 1)² (a) Find the directional derivative at rº = (0, 0) in direction v = (1, 1). (b) Find the best quadratic approximation of the function at xº = (0,0).
Consider the function g defined by 1 g(x, y) = cos (TaV) + log3 (x – y) Do as indicated. In what direction does g have the maximum directional derivative at (r, y) = (4, 1)? What is the maximum directional derivative?

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5) (a) Show that F = (8-14xy2 + 2ye²x) + (e²x - 14x²y) y is conservative. (b) Find a potential function for the field F defined in (a) using the line integral method. (c) Use the "partial integration" method to find a potential function for F, checking that both methods agree.