1 4. Let f(z)=: Show that J(z))dz = 0 , where Cis a simple closed contour not passing throughout origin. But, J is not analytic at the point =0. Does it contradict with Morera Theorem? Explain it.

Q: Use Green's Theorem to find f. [(2x – y+ 4) dx + (5y+ 3x – 6) dy] around a triangle in the xy plane…

A:

Q: (a) Find an equation of the tangent plane to the surface of z = ye(3ry-y²) at (-1, –3, –3). (b)…

A:

Q: 1. Use the tangent plane to the surface f(x, y) = x² + y? mate f(1, 2). at the point P(-1,1,2) to…

A:

Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

Q: The tangent plane to the graph of the function f(x,y)=x+xy+y2, which is parallel to the z=1+2x+3y…

A:

Q: In(xy) cos(Tx + T) when x 2 and y = 1/2.- Find the tangent plane to f(x, y)

A: We have to find the equation of  tangent plane.

Q: Suppose the closed curve C in R? starts at the point (-1,1), goes along the parabola y = x2 to the…

A:

Q: 28. Evaluate by Green's theorem (x - cosh y) dx + (y + sinx) dy, where C is the rectangle with…

A:

Q: rea under the curve y = 2x – x² from x = 1 to x = 2 with n = 4 %3D tonglon

A:

Q: Glven: XY and ZW Intersect at polnt A. Which conJecture Is always true about the glven statement?

A: as given that XY ↔ and ZW↔ intersect at point A. so, here we understand few terms which is given in…

Q: It is easy to see that u(z) = Im (+) is harmonic in the unit disk |2| <1 and lim,→1- u(reio) = 0 for…

A: Given, Uz=Im1+z1-z2,z&lt;1 Now,…

Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, yo, 20) is given by the…

A: Given: Surface f(x, y, z)=c and a point x0, y0, z0 To show: Equation of tangent line to f(x, y, z)=c…

Q: Show that f(z) = x ² + y ² where z = x + iy is not analytic anywhere using Cauchy Riemann equations.

A: Cauchy Riemann Equations: Given,  f(x+iy)=u(x,y)+iv(x,y)  and it should satisfy  ∂u∂x=∂v∂y and…

Q: (7.5.0)- Find an equation of the tangent plane to the surface of z = sin(x – 2y) at the point %3D

A: Given F(x, y, z)=z-sin(x-2y) ∂∂xF(x, y, z)=∂∂x(z-sin(x-2y))=-cos(x-2y)∂∂yF(x, y,…

Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

A:

Q: 4. If f is a differentiable function of a and y with w f(x, y), x =r cos 0 and y = r sin 0, %3D show…

A: To verify the given relations between partial derivatives.

Q: 2. Evaluate directly the line integral fo a dy where C is the quarter-circle from (0, 4) to (4, 0)…

A:

Q: 2. For f(z, y) = z-y, prove that f is differentiable at the point (1, 1).

A: Given: f(x, y) = x2 - y2

Q: 4. Find both points on the surface xz – x2 – y? = 8 where the tangent plane is horizontal.

A:

Q: 1. Find an equation of the tangent plane to the surface of z = sin(x - 2y) at the point (7,2,0).

A:

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

A:

Q: tion z2 – (x2 + y2)2 and let p = (0, 0, 0). ermine the set Tp(Zf ) of all tangent prs to Zf at p. Is…

A: Sol

Q: Show that f(x,y)=exyx is differentiable at point (1,0) using the definition of differentiability.

A: A function is called differentiable, if the derivative of function exists at each and every point in…

Q: 2. Set up the formulation to find the surface generated when the curve y=x is revolved about the…

A: Let f(x) be a nonnegative smooth function over the integral [a,b]. Then the surface area of the…

Q: The equation of the tangent plane to the surface X +y- z = (2, –1, -1) is 2 at the point | O a) None…

A: This question can be solved by applying the formula of equation of tangent plane to the given…

Q: c) Let f(r, y) = Vlry| Prove that f(r, y) is not differentiable at point (0, 0).

A: Given a function : f(x,y)=xy We need to show that f(x ,y) is not differentiable at (0 ,0). Let's…

Q: (6) Suppose the closed curve C in R? starts at the point (-1, 1), goes along the parabola y = r? to…

A:

Q: 5. Are there any points on the graph of x2 - y? – 22 = 1 where the tangent plane is parallel to the…

A:

Q: 1 4. Let f(=) =- s · Show that J (z)dz = 0 , where C is a simple closed contour not passing…

A: Moreras theorem

Q: The tangent plane to the graph of the function f(x,y)=x+xy+y2, which is parallel to the z=1+2x+3y…

A: Here given surface is fx,y=x+xy+y2 or x+xy+y2-z=0......1 Then, consider Fx,y,z=x+xy+y2-z

Q: 22 Find the Directional Derivabive of Pary,z)= xg-y2'+Z at the point CI, -2, 0) in the direcbion of…

A:

Q: (a) Calculate the line integral (4x+2y²)dy where C' is the curve y = 2³, starting from z = 0, y = 0,…

A:

Q: A point moves along the intersection of the elliptic paraboloid z = 4x + 11y and the plane y = 4. At…

A: given information is z=4x2+11y2 The plane y=4To find :Rate of change of z with respect to x when the…

Q: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

A:

Q: Find the tangent plane to the elliptic paraboloid z = x² – 2y at the point (1,1,1). - O A. z = 2x –…

A: Given that: z=x2-2y2      and point is:   (1, 1, 1)

Q: Find the line integral (4х — 5у) ds, where C is the semicircle y = 9 – x² traversed from (3,0) to…

A: Graph the semi-circle. Take x=3cost and y=3sint Now, x'=-3sint and y'=3cost The angle covered by…

Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

A:

Q: 5. Show that the equation of the tangent plane to f(r, y, z) = c at (ro, Yo: 20) is given by the…

A: This is the question from vectors.

Q: Let C be the curve on the xy-plane that is the projection of a horizontal cross-section of the…

A: Given that,

Q: 5. Are there any points on the graph of x2 – y² – 22 = 1 where the tangent plane is parallel to the…

A:

Q: prove that x^2y/(x^2+y^2) is not differentiable at 0,0 when  x^2y/(x^2+y^2) if (x,y) not equal to…

A: f(h,k)-f(0,0) =hfx(0,0) +kfy(0,0) +h2+k2∅(h,k) h2kh2+k2-0 =hA+kB+h2+k2ϕh,k ...(1) A =fx(0,0)…

Q: 3.4.6 Evaluate the line integral [sin y dx + (x- cos y) dy] taken in the anticlockwise sense, where…

A: Green's Theorem States that  Let C be the positively oriented, smooth, and simple closed curve in a…

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: the equation we have  (x3-y4)6+ln(x2+y)=1

Q: *6. Let f: R' →R be given by =+4x'z-6xyz+4y-3x'y. Is M=f-((0}) a smooth surface (2-dimensional…

A: Given, fxyz=z2+4x3z-6xyz+4y3-3x2y2 Now, writing the given function in matrix form.…

Q: 13. Find all points on the surface at which the tangent plane is horizontal. 3,,2 (a) z = x°y ² (b)…

A:

Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

A:

Q: 2. Suppose that a function f: R3 → R³ defined by: f (х, у, 2) 3 (2х + Зу — z, sin(тхy?z-1),…

A:

Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: According to the given information, it is required to use green’s theorem to evaluate the integral.