2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1), Determine its Jacobian Jf at the point (1,1,1).

Q: find the linearization L(x) of ƒ(x) at x = a. ƒ(x) = tan x, a = π

A: The linearization of a function f(x) at x = a is, Now differentiating  given function with respect…

Q: Find the divergence of the field. F = (2x+7y-8z)i + (5x +3y+6z)j + (-6x-y+2z)H div F =

A:

Q: Find the divergence of the vector field F <7", 22°, zy* div F -

A:

Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

A:

Q: V.(V×F)= 0

A: Now given, ∇.∇×F=0 To prove the above.

Q: Evaluate F. dr where F(x, Y, z) = (1.25yz cos(xyz), 1.25xz cos(xyz), 1.25xy cos(xyz)) 1 1 and C is…

A:

Q: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

A:

Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

A:

Q: b) Show the force field M UTM UTM F(r, y, z) = (r²y + In(yz)) i+ UTM 3 + y cos(2z) ) j+ tial…

A:

Q: 7. Find the Jacobian of the transformation T(x, y) = (w sin(v²),w cos(v²)). X=w Sin (v?) w = sin…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question (Q7) for you.…

Q: 5. Let F be the gradient vector field of the function f (x, y) = 4xy + 3y". Evaluate JaF. dr, where…

A:

Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

Q: Find the work done by the force field F in moving an object from P(-3, 7) to Q(7,5). 2x Y F(x, y) =

A: To find the work done by the force field F in moving an object from P-3, 7 to Q7, 5. Fx,…

Q: Let flxiy) = Rx Jy 27 %3D a) Finid the gradrent of f b) Find the directional derivative of f at the…

A: Consider the function as  fx,y=2xy

Q: Find the work done by the force F = x²yi– xy²j along the paths shown from (1, 1) to (4, 2). (4, 2) a…

A:

Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: Consider the vector-valued function [f(x, y) xt – xy² [g(x, y) 2 Vx + y³x² [f(x, y) Which of the…

A: Find the Jacobian of the given matrix

Q: If f(x, y) is a potential function for the gradient vector field F(x, y) (Зх — 2у) і — (2л — у) ј, —…

A: Since F is conservativei.e, ∇×F  , so there exists some potential function f so that ∇f =F .

Q: Suppose that Vf(x, y, z) = 2xyze*i+ ze**j+ ye*´k. If f(0, 0, 0) = 5, find f(1, 1, 8). Hint: As a…

A:

Q: 3 2) Integrarte the vector field hlxry,2)= xyi + ya J+xa k over the twisted cubic PIt)=ti+t?g + t3 k…

A: Given that, the vector field        h(x, y, z) = xy i + yz j + xz k        r(t) = t i + t2 j + t3 k…

Q: Find the work done by the force field F in moving an object from P(-3, 7) to Q(7,5). 2x F(x, y) = -

A:

Q: Find the linearization L(x,y,z) of the f(x,y,z) = 6sqrtx^3+y^3+z^3 at the point (1,2,3).

A: The linearization of a function f(x,y,z) at a point (a,b,c) is given by the expression;…

Q: 1. Use isoclines to sketch the direction field for y' = x+ 2y and sketch the solution curve that…

A:

Q: Find the divergence of the vector field F. F(x, y, 2) = In(9x+ 8y² )i + 72xyj + In(8y² + 5z? )k #6…

A: Solution is done in step 2 and 3

Q: Evaluate both sides for the field- rectangular parallelepiped X = 0 and 1 Y=0 and 2 and Z=0 and 3.…

A:

Q: Is h: (-1,1) → R, h(x) = tan[(t/2)x] homeomorphism? Please search. 14:42 / %3D a

A:

Q: s the linear operator T : R3 --> R3 defined by T (x, y, z) = (2x - 3y + 7z, 5y + z, -z)…

A:

Q: Show that the function f(x + iy) = Vlæ|ly| satisfies the Cauchy-Riemann equations at the origin but…

A:

Q: Let F(x, y) = (4 cos(y), 2 sin(y)). Compute the flux $ F - n ds of F across the boundary of the…

A: By using Green Theorem, ∮F.n ds…

Q: Compute 25 dz, I, (z2 – 3)(z³ + 1)(z+8) where y is defined by z(t) = 2 eit, t e [0, 27]. Hint: in…

A: We will calculate residue at infinity as hint provided.

Q: Find the divergence of the vector field F. F(x, y, z) = In( 7x²+ 4y² )i + 28xyj + In( 4y2 + 5z2 )k

A:

Q: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann equations if ux = v, and u,…

A:

Q: Find the curl of the vectòr field F(x, y, z) = 2xyzi + 8yzj + 5zk

A: According to our guidelines we are supposed to do only one question.  Kindly repost other question…

Q: Express dw/ar and aw/as as functions of r and s using chain rule w = In(x² + y² + z²), x = re sin r,…

A:

Q: 4. let f(x, y) = x² – xy + y² – y. find the direction u and Duf at Po(1,–1) for which Duf is the…

A: Given:- f(x, y) = x2 -xy+y2 -y To find:- Direction u and Du f at point p0 (1, -1) for which Du f is…

Q: Show that the vector field F = 2.xyi +(x² – 2 y)j is conservative and evaluate the line integral…

A: Given the vector field

Q: Use the Divergence Theorem to evaluate the flux of F = yi - xj + 2xk across the sphere x ^ 2 + y ^ 2…

A: Given: Fx,y,z=yi-xj+2xk  we want to calculate flux of Fx,y,z=yi-xj+2xk across the sphere x2+y2+z2=25…

Q: Show that there is no vector field G = (P,Q, R) where P, Q and R have continuous second order…

A: Recall a result, If G= (P,Q,R) is a vector field on IR3 and P, Q and R have continous second order…

Q: 4. Find the work done by the force F(r,y,2) = (x,-z,2y) in moving straight from (0,0,0) to (1.1.0),…

A:

Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

A: Given: The vector field F=Pi+Qj+Rk is conservative P, Q, R have continuous first order partial…

Q: 4. Calculate the flux of F = over the square shown below, oriented counterclockwise. SS (Px +Q, R…

A: We have to check given work on this problem write or wrong   But this work is wrong Because flux or…

Q: Compute the Jacobian of the mapping at the given point. O(r, 0) = (8r cos(0), 8r sin(0)), (r, 0) =…

A:

Q: Show that e", sin x, cos a are linearly independent using Wronskian Method.

A:

Q: Find the Jacobian of following functions f(x,y, z) = 3xy – yz + 2y z2 - 5xyz3 g(x,y, z) = xyz -…

A:

Q: Express dw/dr and dw/as as functions of r and s using chain rule w = In(x + y + z*), x = re sin r, y…

A:

Q: 10. Find the work done by the force F=vx +2y y - 7 for a particle that moves between points (-1,0,1)…

A:

Q: 2: Compute the work done by the force field F = (y, -x) to move an object from point A = (1,0) to B…

A:

Q: B) Find the work done by the conservative vector field F (x, y, z) = yzi+xzj+xyk along a smooth…

A: Since, it is given that the field is conservative.

Q: Justify the following statement: The flux of F = (x3, y3, z3) through every closed surface is…

A: The flux through a closed surface is given by Gauss's divergence theorem, Flux, ∫∫∫∇→·F→dV=∯F→·n^ds…

Types of Bias

Statistics is the study of data collection, organization, analysis, interpretation, and presentation. Statistical bias is a characteristic of a statistical technique or its findings in which the expected value deviates from the actual root quantitative parameter being estimated. According to the actual definition of bias, it refers to the tendency of a statistic to overestimate or underestimate a parameter.