Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) (7 Marks) When: u(x, y) = 2x – x³ + 3xy?

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Q: . Verify that u(x, y) = xy+7x +e° onjugates v(x, y) and find an expression for f(2) = f(x+iy) = u(x,…

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Q: 3. Verify that u(x, y) = x² + 4.x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

A: See the attachment

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Q: - Find div F and curl F if F(x, y, z) = 10y¹z7i – 17xªzºj – 3xyễk. div F= curl F=

A: The general form of a vector-valued function in three dimensions is given by F=F1i+F2j+F3k, where…

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Q: Consider the functions z=7 ex In y, x= In (u cos v), and y= u sin v. dz dz (a) Express and as…

A: Given, z= 7 ex ln y    ....... (1)x= ln ( u cosv) .......... (2)y= u sinv…

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Q: AXYZ is isosceles with base YZ and M is any point on YZ. Prove that a. ZXMY > ZY Z. b. XM < XY M

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Q: Find an antideriuative Flx) of the functian f(x) = 8 sinx such that FlO)=D1 %3D 13 + COSX

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Q: Let z = g (x, y) = x³ + xy³ + y4 - 6x - 2y + 4, where x = r cos (0) and y дz Use the chain rule to…

A: We will use chain rule to find the derivative

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Q: 2 Find.all.se.cond.partial.devevatives:. f(x,y)In(3xi+5.y..b.ue sint C.Z-ytan x a..

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

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Q: Verify that u(x, y) = x² + 4x – y² + 2y is harmonic and find %3D - a) the harmonic conjugate…

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Q: (b) Use Leibniz's product rule to derive the quotient rule y du- v dy d YY

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Q: Verify that u(x, y) = e®²-y´ cos(2xy) is harmonic in R?. Find the harmonic conjugates v of u, and…

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Q: Verify that the real-valued function u(x, y) = x^3 − 3xy^2 − 2y defined on the complex plane C is…

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Q: Let u(x, y) = 2x(c – y), where c is a constant. (a) Prove that u is harmonic on C. (b) Find all…

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Q: 9. Demonstrate that the functions Y1 = x2, y2 = sin(x), y3 = xe are linearly independent on R.

A: Recall: If y1,y2 and y3 are the given function and         Wy1,y2,y3x = y1y2y3y1'y2'y3'y1''y2''y3''…

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Q: Q2 A) for F= 2xi-3Y3+K %3D prove That div( curl È)=0

A: Given :          F→ = 2x i - 3y j + k   To prove that div(curl F→ ) = 0

Q: 2. Find an equation for the plane tangent to the surface 3x2 - 4xy+ z2 = 0 at the point (1, 1, 1).…

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Q: 2- Show that the function U(x,y)= 3x²y-y² + y is Harmonic and then find it's Harmonic conjugate V(x,…

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Q: Show that U (x, y) = e"cos v,V(x, y) = e"sin v are conjugate harmonic of each other if f = u + iv is…

A: Solution given below

Q: Show that u(x, y) is harmonic in some domain and find a harmonic conjugate v(x, y) when (a) u(x, y)…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Prove that arctan x + arctan y = arctan(x + y) / (1 − xy),  xy ≠ 1.                   Use the…

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Q: Q14] Verify that u = xy – x + y is harmonic and then find a conjugate (v) of u. Suppose that v (0,…

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Q: Show that u (x, y) is harmonic in some domain and |find a harmonic conjugate v(x, y) when y (a) u(x,…

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Q: =y一;-T十;-一

A: Draw the family of these one parameter DE's of bifurcation.    Further, to solve these.

Q: 2. Find the directional drevative of f(x, y,z) xsinyz at the point (1,3,0) in the direction of i =i+…

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Q: Show that the graph of ř = (t sin t, t cos t, t²) lies on the paraboloid z = x? + y?.

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Q: Consider the functions z = - e* In y, x= In (u cos v), and y = u sin v. dz dz (a) Express du and dv…

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Q: If F (2x + y)i + (3y 4x)j evaluate o. F.dr around a triangle ABC in the xy-plane with A(0. 0). B(2.…

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Q: Q5: Find W,e if W = f(x,y), xr cos6 and y =r sino, then prove that r Wre =-yf + xfy+ xy(fyy-fex) +…

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Q: Show that the subtangent to y = x' at x = a is equal to a/3.

A: Definition used -                            Length of subtangent =ydxdy