Test for Determining if a 3-D Vector Field is Conservative: Let M, N, and P all have continuous first derivatives in an open region R. The vector field given by: F(x, y, z) = (M,N,P) is conservative if and only if: ӘР ƏN әм ӘР ƏN дм " , ду Əz Əz дх дх ду DEFINITIONS Any expression M(x, y, z)dx + N(x,y,z)dy + P(x, y, z)dz is a differential form. A differential form Mdx + Ndy + Pdz is EXACT on an open, simply connected domain D in space if and only if 224

Sketch the vector field F(x,y)=1/2xi-1/2yj at points (-1,1), (-1,0),(-1,1),(0,-1),(0,0),(0,1),(1,-1),(1,0),(1,1)

V=(y,2y,0) and w=(x-y,x+y,0)

2. For the vector field F (x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div (F)div (F) (b) curl (F)curl(F) (c) div (curl (F))

2. Consider the vector field F(x, y, z) = 5zi - 7y3j+5xzk. (a) Calculate div F at the point (1,1,0). (b) Explain the meaning of the number you found above.

What are V-F and VX F for the vector field F(x,y,z) = 3x²zi-ye*j - y²zk? OV.F = x²exy 2 VXF = x²i-3e²j+yk OF=3x²-xe* + 2yz VXF = 2i-3j-k V-F = 3x²-x-z VXF= -2yi+ 3xj + zk V-F = 6xz-e* - y² VXF = -2yzi + 3x²j-ye*k

Sketch the vector field F(x,y) → xi + 3yi - a. C. 'ீ, -10 10 -10 -5- L -10- 10- -10- 10 A R -10 10 10 10 b. d. -10 -10 10- IDA -10- וווו -10 10 10 d ரு VD

7.Vector analysis (calculus 4)

Let F(r, y) be the vector field plotted below, and let C be the piecewise-linear path from (8, -5) to (8, 5) to (10, 5). What is the sign of / I F. dr? 10 -10 -10 10 Select one: a. positive O b. negative C. zero

We are given a vector field and a parametric curve