What does it mean for the differentiability of a function if only one of the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) holds?

Q: 1-Find the gradient of (1) f (x,y)= In(x²+y²) (1,1)

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Q: Find the linearization at x = a. IT f(x) = sin³ (x), a = 4

A: Given,            The linearization at x=a.             fx=sin3x, a=π4We know that,             The…

Q: Prove that = va² – x* if y =(f)sin~+ () + (9)v&* - x²

A: Differentiation wrt x

Q: Which is the linear differantial equation into which the Bernolli differantial equation transforms

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Q: Determine whether the functions f1(r) = e2*, f2(x) = e-2" , and f3(x) = sinh 2x are linearly…

A: To determine whether the functions f1x=e2x, f2x=e-2x, and f3x=sinh 2x are linearly independent on…

Q: the linearization of (fi,f2) = (x²y+ 3xy², 3x³ + 2xy) at (1,0) to approximate the value at…

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Q: A:- Find the orthogonal trajectories for functions In -1) = - COS X +a y

A: The given family of curves is: lny-1y=-cos x+a ,            where a is an arbitrary constant.

Q: Find the derivatie: F (+2 %3D (-t*+*z+)た =人

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Q: 8. The curl of gradient of a scalar function is (a) zero (c) unity (b) non-zero (d) depends on the…

A: the answer is option a.) zero the explanation is given in the below step

Q: Determine the orthogonal trajectories of the family of electric field lines described by the family…

A: Given    x-c3+y2-c23x-c=x3   differentiate  w.r.t.x    3x-c2+2ydydx-3c2=3x2⇒2ydydx=3x2-3x-c2+3c2…

Q: 2. Find the gradient vector at the indicated point f(x, у, 2) 3 ху — In (2),. (2, -2,2)

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Q: (6) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

A: Orthogonal trajectory curve of a family of curve is defined as a curve which intersects every member…

Q: 8. What is the relationship between a and b so that F(x, y) = (ary, 3ax + bry) is source free?

A: That's easy. Have a great day!!!

Q: Find the linearization of z = f(x,y) at P(-3, 2) when f(-3, 2) = 2 and f-(-3, 2) = 1, fy(-3, 2) =…

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Q: Find the linearization of the function a) z = cos(sin y – x); at (-2,0) and use it to approximate…

A: Suppose f(x,y) be a differentiable function at (a,b). Then linearization of f(x,y) at (a,b) is given…

Q: 2. Show that the function u p(xy) + Vry (y/x) provides the equation = 0.

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Q: The gradient vector of the function f(x, y) = cos(4x - 6y) is (4 sin(4x - 6y), -6 sin(4x - 6y)).…

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Q: Let A = {(x,y) = RXR; 0<max{|x-3], ly- 2|}<1}, then: *

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Q: 4. Use the linearization of f(x, y) Vx + 2y to approximate f(3.01,2

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Q: Use the linearization L(3,-2) (x, y) of f(x, y) = -x²y-xy² near the point (3, -2) to approximate the…

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Q: The gradient vector of the function f(x, y) = sin(4x - 6y) is (4 cos(4x - 6y), 6 cos(4x - 6y)).…

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Q: Find the orthogonal trajectories of the family of circles passing through the points (1, −2) and (1,…

A: Given that

Q: Find du -if v = t- dt

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Q: What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy…

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Q: Find the best linear approximation, g, to f on the interval [0, 1]. (x) = 2/x g(x) %3D

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Q: Find the orthogonal trajectories of the family of x² – y? = cx = CX -

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Q: Verify Cauchy-Riemann equation for the function f(z) = z^3

A: To find- Verify Cauchy-Riemann equation for the function fz = z3         Formula Used-…

Q: what are the independent and dependent variables of this problem? is the DE separable?

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Q: Find the linearization of the function at x = 1. * f (x) = 2x³ - 3x + 7 O y = 6 + 1(x - 3) O y = 6 +…

A: Given, fx=2x3-3x+7

Q: Find the linearization of f(x,y) at the indicated point (X0,Yo): f(x,y) = In (x² + y); (1, 1) L(x,y)…

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Q: Find the best linear approximation to f(x) = ex on the interval [-1,1]

A: We have to find the best linear approximation of f(x)=ex on the interval -1,1.

Q: Choose the gradient of the function f defined by f(x, y, z)= In[1 + (7xy + 13xz + 17yz)²]. O…

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Q: Find the derivative of f(z) = In(z -+ O (A) f'(2) = 2² – 3z + 2i). | --2 V:-3:+2; 1 AB) ƒ'(2) = V-3:…

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Q: se Cauchy-iemman equations to now that each of these functions is where cnalytic Rz= メ

A: Given: fz=xy+iy   -1 The objective is to show that the given function is nowhere analytic.

Q: 2. Show that cos(wt – B), coswt, sinwt are linearly dependent functions of t. Show Complete Solution

A: Given that: cos(wt-B), coswt, sinwt

Q: Find the orthogonal trajectories of У — сх CX

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Q: 7 For the function eX, the linear approximation around x = 2 is %3D

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Q: Show how the real and imaginary parts of the function satify the Cauchy-Riemann equations. f(z) =…

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Q: Find the orthogonal trajectories of x^2+y^2−2cy=0

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Q: Find the linearization of the function f (x) = a° at a = 1.

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Q: 4)Find the scalar function ø such that F = V ø where F=- y * +y x +y

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Q: Show that the linearization of f(x) = (1 + x)k at x = 0 is L(x) = 1+ kx.

A: We need a linear approximation formula for f(x).

Q: What is dufferential equation of x'=4x-y and y'=-4x+4y when t=0, x(0)=1 and y(0)=1

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Q: Integrate the function f(x, y) = x – y over the triangle T with vertices (0,0), (1,0), and (1, 2).

A: Evaluate : ∫∫T(x-y)dA        ......(1) ∫∫Tx-ydydx T is the region enclosed by the triangle. Region…

Q: Show that the functions ƒ 1 (x) = sin x, ƒ 2(x) : = cos x, and f 3(x)=x cos x are linearly…

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Q: 1. How do you find the linearization of f(x) at x = a? Illustrate with a concrete example

A: We have to solve

Q: (6) Find the orthogonal trajectories of the family of curves r= 2C sin 0, where C being parameter.

A: See the detailed solution below.

Q: Find f (1, 1) S(x, y) = In( %3D Vx² + y²

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.