Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that |f' (z) | <=  1/1-|z|

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

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Q: Find the divergence of the field. F=(-x+8y - 3z)i+ (8x- 4y + 3z)j+ (- 4x- y- 3z)k

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Q: A vector field is described as F = (-z + 7x, x³y²z=1,3x + 2z²). (i) Calculate curl F. (ii) Determine…

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Q: Consider vector field: F = e i-cos y j+ sin² z k. Obtain divergence and curl for F at points (0,,).

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Q: If a vector field F (x, y, z) = (evz-, In (x² z), sin z), which of the following values is equal to…

A: We have to find div F(-1,0,1).

Q: 5. Let F(x, y) = ei + xej be defined on a smooth curve C given by the vec function r = ti+t2j for 0…

A: A detailed solution is given below

Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: What is the divergence of the vector field f = x²i+ xyzj – zk at the point (1, 2, 3). Select one: а.…

A: f=x2i+xyz j-zK at (1,2,3)  div f=i ∂f∂x+j.∂f∂y+K.∂f∂z

Q: Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

A: This question is belong to the vector calculus in which we have to deal with the divergence theorem…

Q: 3 , find a potential function for the vector field F by inspection or show that one does not exist.…

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Q: 1. Give an example of a vector field F(x, y) in 2-space such that F has constant direction but || F…

A: The solution are next step

Q: Use Divergence Theorem to find the outward flux of field F = (-5x³ – 5y)i + (yz + 1)j + xyzk through…

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Q: Find the divergence of the vector field F = div F = %3D

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Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

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Q: Q3: Prove that the divergence of a curl is always zero.

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Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

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Q: Consider the vector field F = 5 sin(x² )ï + (3x² + 2e* )j. Find the line integral F· dr, where C is…

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

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Q: b) Evaluate both sides of the divergence theorem for the vector field D = 2xyx + x²9 [C/m²] and a…

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Q: 9. Show that F is a conservative vector field and use this fact to evaluate f F dr along the given…

A: Given information: Fx,y=y2i+x2j C is the arc of the parabola y=2x2 from (-1,2) to (2,8). Gradient…

Q: Find the tangent plane for the level surface f(x,y,z) = 3x5 + cos (5y) + In (5z) = c at the point…

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Q: Given the vector field F(x,y,z) = [x, y, -2z] satisfies div(F) = 0. Find a vector field G(x,y,z)…

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Q: find the divergence of the field. 1. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)k 2. F = (x ln…

A: Consider the vector field Substitute the value of vector fields then Hence, the divergence of…

Q: Calculate the divergence of the scalar field F(x,y)=3x2+siny.

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Q: Find the divergence of the field F. F = 8x 5 i + 3y 5 j + 2z 5 k 8x4 + 3y4 + 2z4 O 40x4 + 15y4 +…

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Q: Show that F is a conservative vector field and use this fact to evaluate  F. dr along the given…

A: For a vector field of the form F=Pi^+Qj^ to be a conservative vector field, it should satisfy the…

Q: find the work done by the given vector field F along the specified curve C.   F(x, y) = yi − x j,…

A: Given, F(x,y)=yi^-xj^ where curve, x2+y2=4 W=∫F.dr W=∫ydx-xdy from curve, dx=-dy W=∫xdx+ydy limit of…

Q: Let C be any smooth simple closed curve with positive orientation that lies entirely in a plane with…

A: Step:-1 Stock's Theorem:- ∫CF.dr=∬Scurl(F). ds=∬Scurl(F). n dA Part (a):- Given that F→=9y, 3x, 6z…

Q: Find the divergence of the vector field F div F -6z sin (a + 7) %3D

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: 6. Find the circulation of the field F(x, y) = -yi + x²j along the closed elliptical path x² + = 1…

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Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: 3. Find a field F(r,y) (M(r, y), N(r, y)) in the xy-plane with the property that at each point (r.…

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Q: Find the divergence of the field. F = (- 6x + 8y – 4z)i + (- 8x – 5y + 3z)j+ (x + 5y + 2z)k div F =

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Q: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

Q: 1: Let Let F = (yz², tan-1(xy), 1 + xy³). %3D a) Show that F is not conservative. b) Show that F…

A: conservative vector field is a vector field that is the gradient of function

Q: Consider the surface integral of the vector field F= [xy², 3x², x²z]over a surface S, where S…

A: From Gauss divergence theorem, we have ∬SF→⋅dS=∭VdivF→dV   The vector field is F→=xy2,3x2,x2z.…

Q: (a) For any vector field F, the gradient of the divergence of F is always zero.

A: Since the gradient of divergence of F is given by

Q: Find the divergence of the vector field F. F(x, y, z) = In 5x²+ + 10xyj + In( 2y2 + 722)k div F(x,…

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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: 3. Use the divergence of the vector field to compute F(x, y) = (tan-1 (2x)) ₁ + (xy-2 +²8²) J j 4 fo…

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Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

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Q: 4- Evaluate the surface integral of the vector field F = = [ax, by, cz] over the sphere a2 + y² + z²…

A: From the given problem:   F→=axi+byj+czk   Sphere:  x2+y2+z2=36.   From the divergence theorem:…

Q: P=(x²+y²).x+ (xy-y²).ý

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Q: Compute the divergence and curl of F= -(z³-3x)j+4y²k+x²yi

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Q: Use the divergence of the vector field to compute F(x, y) У · (tan-¹ (2x))i + (xy − 2 + 8x²) j - = 4…

A: we are given the vector field as  F(x,y)=tan-1(2x)4i+(xy-y2+8x2)j to compute…

Q: Give an example of a vector field F (x, y, z) that has value 0 on precisely one line and such that…

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Q: Find the divergence of the vector field F %3D div F =

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Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution