Let f(2) = Re(z) be a complex valued function defined on a complex plane С. Where is f differentiable?

Q: Find and classify all stationary points of the function f(x,y) = x³ + xy + y² − x.

A:

Q: Is the function eiz² holomorphic in C? Explain why. What are the Cauchy-Riemann equations for eiz?

A: Given function : f(z) = ei z2

Q: analytic function a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all o. #2.…

A: If f(z)=u+iv is analytic in a domain D , then f satiesfies the Cauchy Riemann (C-R) equations that…

Q: Verify that U(x,y) = x3y - xy3 is harmonic on the complex plane.

A:

Q: 2. Consider the function f(z) = where z is a complex variable defined as z = x + jy. nvestigate…

A: Given: The given function is fz=1z we want to check that whether given function is analytic or not…

Q: Calculate the gradient vector Vf of the function f(x, y) = x² – x + y- x²y – 2y2 at the point (2,1)…

A: According to the given information, it is required to determine the gradient vector of the given…

Q: What is the graph of r(t)=2 costi + sintj?

A: Vector-valued function : It is a function of the form  r(t)=f(t)i⏜+g(t)j⏜ or…

Q: Show that ƒ(z) = 3x² + 2x − 3y² − 1 +i (6xy + 2y) is differentiable at every point in the plan -

A: The given function is f(z)=3x2+2x-3y2-1+i(6xy+2y)...(i).

Q: (b) part of parabola y = −2x² from (−1, −2) to (2, —–8)

A: Given Equation of parabola y=-2x2 from -1, -2 to 2, -8.

Q: Verify that the function f(z) = Z is continuous throughout the complex plane.

A:

Q: Let r(t) be a vector-valued function such that the magnitude of r(t) does not change over time. Use…

A: if r'(t) = 0 then we can say that r'(t) is perpendicular to the function r(t) for all times t.

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: Write the function f(x) = √1 + x² as a polar function.

A: Let

Q: What is the geometrical meaning of an integral of a vector function?

A: We can realize geometric meaning of integration such as

Q: Let ƒ(z) = |z|², z € C. Show that f is complex-differentiable at z = 0, but nowhere else.

A:

Q: An analytic function w = f (z) maps the second quadrant of the z-plane to the interior of a unit…

A:

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.