MA330 Homework #4 1. This exercise concerns the function its gradient vector field F-vo See the p...
(1 point) Math 215 Homework homework9, Problem 2 Find the gradient vector field of the function f(x, y) = -75x2 + y2. F(x,y) =
Consider the following potential function. a. Find the associated gradient field F =Vo. b. Sketch three equipotential curves of Q. c. Show that the vector field F is orthogonal to the equipotential curve at all points (x, y). 5) (12 points) $(x, y) = 2x² + 2y2
Prove that the following vector field F = 4xi +z j +(y – 2z)k is a gradient field, which means F is a conservative field and the work of F is path independent? Show all your work. a) Find f(x,y,z) whose gradient is equal to F. Is the line integral ſi. · di path independent? b) Find the line integral, or work of the force F along any trajectory from point Q:(-10, 2,5) to point P: (7,-3, 12).
please answer asap (it is all the professor asked) (5) Consider the gradient vector field F ▽f where f(x,y) = cos(2x-3y). Find curves G and C2 that are not closed such that JG F·dr = 0 and 1, F . dr-1. Explain why you pick the curve you do, and how you know the integrals have the correct values. (Hint: Try picking a straight line between the origin and some simple point (a, b) that you choose later.) (5) Consider...
(1 point) (a) Show that each of the vector fields F = 4yi + 4xj, G= x y zit vol y J, and ] = vertinant virtuaj are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y) = For i, a potential function is h(x, y) = (b) Find the line integrals of F,...
we need to determine if the vector field depicted in graph 1 and graph 2 are conservative by using the last 3 bullets points in the picture Project 1. Fundamental theorem of line integrals In our course we learned the fundamental theorem of line integrals: if F is a conservative vector field with potential f and C is a curve connecting point A to b, then f-dr = f(B)-f(A). Moreover it happens if and only if for any closed curve...
Line Integral & Path Independency Problem 1 Prove that the vector field = (2x-3yz)i +(2-3x-2) 1-6xyzk is the gradient of a scalar function f(x,y,z). Hint: find the curl of F, is it a zero vector? Integrate and find f(x,y,z), called a potential, like from potential energy? Show all your work, Then, use f(x,y,z) to compute the line integral, or work of the force F: Work of F= di from A:(-1,0, 2) to B:(3,-4,0) along any curve that goes from A...
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient field and then find the function f(x,y) such that F = VS. Use it to evaluate line integral SF. dr where the curve C is the arc of the circle 12 + y2 = 4 from (2,0) to (0,2)
(1 point) (a) Show that each of the vector fields F-4yi + 4x j, G-i ЗУ x2+y2 x?+yi J, and j are gradient vector fields on some domain (not necessarily the whole plane) x2+y2 by finding a potential function for each. For F, a potential function is f(x, y) - For G, a potential function is g(x, y) - For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C...