Determine whether each of the following vector fields F is path independent (conservative) or not. If it is path independent, enter a potential function for it, that is, a function f(z, y) so that Vf = F. If it is path dependent, enter NONE. (a) If F(x, y) = (x + 2)i + (3y + 4)j, then f(x, y) = (b) If F(x, y) = (4x + 5y)i + (5x + 2y)j, then f(x, y) = (c) If F(x, y) = (-2y, 2æ), then f(x, y) =

Q: 3. Calculate the directional derivative of the function f : R³ → R, defined by f(x, y, z) = y²+xz at…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1. Determine the domain and range of the vector-valued function r(t) = (√t+3, In(2 – t), t²)

A:

Q: Determine whether the vector functions, x1(t)= [ et ]           [ -et ] and x2(t)= [ et ]…

A:

Q: 1. Consider the pointsS= (1,−3) andT= (2,5).(a) Sketch the vector−→ST.(b) Give the component form…

A: Here 2 different problems are provided. As per our Bartleby Guidelines, we are supposed to…

Q: Let r(t) = 5e7t + e2t and y(t) = 3e2t + 1le (i) Rewrite the given functions in the vector form Y(t)…

A:

Q: (c) If f is a C² scalar function, then V × (Vƒ) = 0. True False (d) The lines 1₁ (t) = (1 − t, 1+ t,…

A: (C) If c is scalar function Means      curl(grad ƒ)  is = 0   Therefore correct ans is  True

Q: Sketch the vector fields. Use a table for it.    F(x,y)=<x,y-x

A: Sketch the vector field Make a table to sketch the vector field. Table should have values (x,y)

Q: 2. Determine whether the following vectors are linearly dependent in R³: (O)--- ()--C) 1 , = , ez =…

A:

Q: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . Verify that x − p is orthogonal to p.

A:

Q: Find a vector-valued function whose graph is the plane x+y+z== a. r(u,v)=ui-vj-(3-u-v)k b. r(u,v)=…

A: The equation of plane x+y+z=3

Q: show that the given vector functions are linearly independent on (−∞,∞).

A: To show that the given vector functions are linearly independent x1t=   et-et,        x2t=etet

Q: 2. (а) ở and w are vectors in R? such that f(0) = w. Express f(27 – 3w) as a linear combination of i…

A:

Q: If the following vector function are equal to: A(t) = 5*t i + 2*t^2 j B(t) = 3*t^2 + 2 j Then…

A: The given problem is to find the derivative of the dot product of the two vector functions. Given…

Q: Compute the first two steps of the Jacobi

A: Given,          2-10-12-10-12uvw=020

Q: Let u = (-1,0,1), v= (3,2, 1) and w = (0,1, 2) . (a) Prove that the vectors u – v, u+ v and u x v…

A: Given : u=&lt;-1,0,1&gt; v=&lt;3,2,1&gt; w=&lt;0,1,2&gt;

Q: 4) Come up with two vector functions r₁(t) and r₂(t) that are non-linear and satisfy the following:…

A: