(1 point) lf F is a path-independent vector field, with Г F-dr-4.5 and Æ-3 and (0,0) F d -3.7, find (0,0) dr (1 point) lf F is a path-independent vector field, with Г F-dr-4.5 and Æ-3 and (0,0)...
4. Let F(x,y) - PiQj be a smooth plane vector field defined for (x,y) f (0,0), and F - dr for integer j, and all suppose Q - Py for (z, y) (0,0). In the following L-JF dr for integer j, and all G are positively oriented circles. Suppose h = π where G is the circle x2 + y2-1. (a) Find 12 for G : (x-2)2 + y-1. Explain briefly. (b) Find Is for Cs: ( -2)y 9. Explain...
Find the work done by the vector field F(x, y) = {xy i + áraj (the vector field from Question 1) on a particle that moves from (0,0) to (0, 1) (moving in a straight line up and along the y axis) and then from (0, 1) to (3, 2) along the curvey= Vx+1. Thus the path is given by along the curve y=x+1 (0,0) up the y-axis + (0,1) (3,2) 1 F. dr 2 F. dr = 0 18...
3) Given vector field F(x,y,z)=<y, xz,x? >. Find N dr where T is the path around the triangle with vertices (1,0,0),(0,1,0) and (0,0,1) traced counterclockwise (when viewed from above.)
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...
Please do the parts in the given order tyā (x,y)メ(0,0) (x,y)= (0,0). if if 1 (d) Given the unit vector u-( find the directional derivative of f(x, y) at the 리지, ,- point (to,m) = (0,0), in the direction of the vector a. (e) Find the gradient of f(x, y) at the point (zo,o) (0,0) (c) Find the equation of the tangent plane to the graph of the function z -f(x, y) at the point (x,y,z) (1,0,0). tyā (x,y)メ(0,0) (x,y)=...
Let F = (P,Q) be the vector field defined by P(x,y) ity, (1, y) = (0,0) 10, (x,y) = (0,0) Q(x, y) = -Ity. (x, y) = (0,0) 10, (x, y) = (0,0). (a) (3 points) Show that F is a gradient vector field in RP \ {y = 0}. (b) (4 points) Letting D = {2:2020 + y2020 < 1}, compute the line integral Sap P dx +Qdy in the counter-clockwise direction. (c) (1 point) Does your calculation in...
4. Stoke's Theorem: Consider a vector field F = (1,1)+(1,0) + (0,0). tyle +rin the unit square bordered by (0,0) + (0,1) ► (a) What is the curl of the vector field F? [2 points) (b) What is the path integral of the vector field around the unit square? [5 points) (c) Show your answers to the previous parts are consistent with Stoke's Theorem. HINT: consider the right-hand rule. (3 points]
For a vector field F(x)(2yarctanx)j find a function f such that F,y)-V/ h(2yarctanx)j find a function f such that F(x,y)-U For a vector field F(x,y)- 1+x2 and use this result to evaluate dr, where C: rit2, osis1 For a vector field F(x)(2yarctanx)j find a function f such that F,y)-V/ h(2yarctanx)j find a function f such that F(x,y)-U For a vector field F(x,y)- 1+x2 and use this result to evaluate dr, where C: rit2, osis1
True or False Determine whet her the statement is true or false, and circle the correct answer. Each question is worth 2 points. (1) If F is a vector field and C is an oriented curve, then F dr must be less than zero. F (2) It is possible that for a certain vector field F and piecewise smooth oriented path C we have/. F. dr-2i-Sj. (3) Suppose d·is the unit square joining the points (0,0), (1,0), (1,1), (0.1) oriented...
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).