What does it mean for the differentiability of a function if the Cauchy-Reimann equations (Ux = Vy and Vx = -Uy) do not hold?

Q: EX Prove that the directional derivative of f(x, y) = y² + 2xy³ + x²y² at point (0, 1) in the…

A: What is Directional Derivative: The directional derivative of a scalar function represents the rate…

Q: What is the directional derivative of f(x, y, z) = bx²y²z² + x² + y² + z² in the direction of v =…

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Q: 2. Find the directional derivative of o = yx2 +x²z + 3xyz in the direction of vector n = î + 2 ŷ -2·

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Q: Find the gradient of the function at the given point. In(x2 - Y) - 9, (3, 8) Vz(3, 8) =

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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: What is the formula for the linearization of a function f at the point x = a? %3D

A: Given :To provide the formula for linearization of a function f at the point x=a

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

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Q: 3. Prove that the directional derivative of f(x, y) = y² + 2xy³ + x²y² at point (0, 1) in the…

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Q: Find the gradient of the function at the given point. In(x2 - y) - 2, (2, 3) Z = Vz(2, 3) =

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Q: are functionally dependent, (i) Verify the variables x and y are given by x = and obtain the…

A: Here, the given functions are: x=s+ts=1+ts,y=s+tt=st+1 To verify the variables x and y are…

Q: Which of the following is the gradient vector of the function f(x, y) = 3x ln 3y - 2x²y at point (1,…

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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: 11. Verify that the conclusion of Clairaut's Theorem holds for the function: f(x, y) = e*Y sin(y)

A: The given function is: fx,y=exy siny To verify that the conclusion of Clairaut's theorem hold for…

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. Зx +…

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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: What is the gradient of the function f(x) = 5 sin(5x) A sin(5x) - sin(5x) E B 25 cos(5x) F -cos(5x)…

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Q: Solve ux – uy +yu = 0, where u is a function of x and y. %3D

A: Given: ux-uy+yu=0. The objective is to solve u, where u is a function of x and y.

Q: Find the gradient of the function at the given point. In(x² – y) - 9, (3, 8) z = Vz(3, 8) =

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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: 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: 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: 6) If y=x² +1+ f dt , prove that y"= 2 - x2

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Q: What is the gradient of f=x^2+y^2-2xy-x+3y-z=-4 ?

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Q: Show that the function defined by xy √x²+y² f(x, y) {{ = 0 if (x, y) = (0,0) is not differentiable…

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Q: Find the gradient of cos(xy) – 3y4 + 3 at the point (1,4) -

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Q: What does it mean for the differentiability of a function if only one of the Cauchy-Reimann…

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Q: Find the linearization of f (x, y)= e²"¬*, at (0, 7 / 2).

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Q: Find the gradient function for: 1+3x 1) y = %3D x2 +1

A: Given that  y=(1+3x)/(x^2+1) To find the gradient of y

Q: 2 - What is the directional derivative of the function f(x,y,z) = x3 - 2xy +z at point Ao= (1,1,0)…

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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: w = 3x² – 5y² + 2z², (1,1, –2)

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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 gradient of w= 3x² - 5y* + 2z at the point (1, 1, -2)

A: The given function is: w=3x2-5y2+2z2

Q: First, compute the gradient of the function f(x,y) = 4+5x²-3y². Then evaluate it at the point…

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

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Q: What is the linear approximation for any generic linear function y=mx+b?

A: Given: y=mx+b

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

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Q: 3. Let f(x, y) = In 2x + 4y, find the directional derivative of f(x, y) in the direction of at the…

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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: Find the gradient of the function at the given point. In(x2 - у) - 3, (2, э) Vz(2, 3) = %3D

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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: Compute the gradient for the function f (x, y) = 4x² – 9y at the point (10, -9) %3D

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Q: Find the direotional derivative of the scalar function f-x'+ y' +2z', at the given point (1,1,2), in…

A: Explanation of the answer is as follows

Q: Find the orthogonal trajectories of the given family of curves x^2 + y^2 = c^2. *

A: Curves:  x²+y²=c²

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.