(a) Consider the analytic function f(z) = x − 2x 2 2y 2 i(y − 4xy) This function defines two families of level curves u(x, y) = c 1 and v(x, y) = c 2 Sketch or plot these two families of level curves on the same axes (See for example Slides 40 and 41 in the Differentiation notes) Verify that the families are orthogonalGiving us 1 2 (1−cos(1))≤S 1 0 S 0 sin(x) 1(xy)4 dxdy≤1 5512 Find the volume of the solid bounded by x2 2y2 =2;z=0;and xy2z=2 x2 2y2 =2 de nes a vertical cylinder that crosses the xyplane in an ellipse z=0 is the xyplane Since the ellipse in the xyplane given by x2 2y2 =2 does not intersect the line xy=2, the plane xy2z=2 crosses the cylinderFrom Equation (2118), dy/dx = v/u = x/y, and vdxudy=0 (2118) y dy = x dx Integrating, we obtain y^{2} = x^{2} c where c is a constant of integration For the streamline through (0, 5), we have 5^{2}=0c or c=25 Thus, the equation of the streamline is x^{2} y^{2} = 25
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U-v=(x-y)(x^2+4xy+y^2)
U-v=(x-y)(x^2+4xy+y^2)-1 Answer to Given (uv) =(xy)Find dy/dx x^24xyy^2=4 Differentiate both sides of the equation Differentiate the left side of the equation Tap for more steps Differentiate Tap for more steps By the Sum Rule, the derivative of with respect to is Differentiate using the Power Rule which states that is where
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Find the analytic function f (z)=uiv in terms of z if uv= (xy) (x2 4xy y2) 0 41k views Find the analytic function f (z)=uiv in terms of z if uv= (xy) (x2 4xy y2) written 12 months ago by teamques10 ★ 25k • modified 12 months agoSolutionSolution Let~r(u;v) = hx(u;v);y(u;v);z(u;v)i Note that~r u= h1;1;0iand r v= h1;JNTU BTech M2 Maths and Bsc Maths Chapter Partial Differentiation Topic Maximum and MinimumExamine the function f(x ,y)=x^4y^42x ^24xy2y^2 for ext
2i Hence ~r u ~r v is the downward normal and F~ (~r u ~r v) = D x;2i= x(u;v) 1 y(u;v) = 2u 1 Thus ZZ S F~ dS~ = Z 4 0 Z 2 0 (2u 1)dvdu = Z 4 0 (2u 1)v 2 0 du= 2 Z 4 0 (2u 1)du= 2 Solved (2y^{2} 2xy 3x)dx (y 4xy x^{2})dy = 0
@v @x = 2x 6y 2 Both Cauchy Riemann equations hold at all points and thus f 1(z) is analytic in C The analytic function f 1(z) is a polynomial of degree 2 and we can express in terms of zby considering a nite Maclaurin series representation We have f 1(0) = 0 f0 1 v y = u x = 3 x 2 2 y − 4 y 2 and after integration v ( x, y) = 3 x 2 y y 2 − ( 4 / 3) y 3 β ( X) and after trying to solve for β I found it equal to β = x 2 y − x 2 and after applying it to the v ( x, y) = 4 x 2 y 2 = ( 4 / 3) y 3 − x 2 which in obviously is not applying CR equations if we want to prove the solution2 MAT 436/536 FUNCTIONS OF A COMPLEX VARIABLE HOMEWORK 3 (3) Suppose that f (z) = x2 y2 2y i(2x 2xy), where z = x iyUse the expressions x = z ¯z 2 and y = z z¯ 2i to write f (z) in terms of z, simplify the result Solution f (z) = x2 y2 2y i(2x 2xy) = x2 2xyi y2 2y i2x = (x iy)2 2i(x iy)= z¯2 2iz (4) Show whether or not f (z) = Rez = x is analytic Solution u(x, y) = x and
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For a calculusfree solution, start with the factorization 2x2 −3xy− 2y2 = (x−2y)(2x y) and make the substitution u = x−2y, v = 2xy Then the condition 25x2 −xy 40y2 = 36 turns 2x2 −4xy −6y2 https//wwwtigeralgebracom/drill/2x~24xy6y~2/A steady incompressible flow field is given by u = 2x2 y2 and v = 4xy The convective acceleration, along x direction at point (1, 2) is Q6 Water flows through a pipe with a velocity given by V → = ( 4 t x y) j ^ m / s, where ĵ is the unit vector in the y direction, t (>0) is in second, and x and y are in metersThey are "First Order" when there is only dy dx, not d 2 y dx 2 or d 3 y dx 3 etc Linear A first order differential equation is linear when it can be made to look like this dy dx P(x)y = Q(x) Where P(x) and Q(x) are functions of x To solve it there is a special method We invent two new functions of x, call them u and v, and say that y=uv
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Stepbystep explanation Here, we have analytic function f (z)= uiv and given the real part value, u = (x cos y y sin y) (1) so, by the Cauchy Equations Partially Differentiating Eq (1) with respect to x, we get (2) Partially Differentiating Eq (1) with respect to y, we get (3) Eq (3) equals to the or1 A harmonic function is analytic if it satisfies the Laplace equation If 𝑢 (𝑥,𝑦) = 2𝑥2 −2𝑦2 4𝑥𝑦 is a harmonic function, then its conjugate harmonic function 𝑣 (𝑥,𝑦) is (A) 4𝑥𝑦 −2𝑥 2 2𝑦 2 constant (B) 4𝑦 2 −4𝑥𝑦 constant 2𝑥 2 −2𝑦 2 𝑥𝑦 constant (D) −4𝑥𝑦 2𝑦 2 −2𝑥 2 constant Answer GATE 17 Set2 1 Find an analytic function whose real part is u = x^3 3xy^2 3x^2 3y^2 3y^2 1 asked in Mathematics by Sabhya (712k points) analytic functions;
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The answer is dy/dx=(2xyy^2)/(x^22xy) We use the product rule for differentiation (uv)'=u'vuv' (x^2y)'=2xyx^2dy/dx (y^2x)'=y^22xydy/dx (2)'=0 Putting it allSolution of Bs Grewal higher mathematics engineering book btech Solutionlet the equation of a sphere of the radius isx^2y^2z^2=01 eqn Let x,y,z be the discussion of the rectangular in sphere 1eqn, then, its volume,Using quadratic interpolation find In 92 from In , In and In A The given problem is find the approximate value of ln(92) from the given data by using quadratic Q Convert the following triple integrals to cylindrical coordinates or spherical coordinates, then
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Then, in general, it is not possibleUNIT 148 PARTIAL DIFFERENTIATION 8 DEPENDENT AND INDEPENDENT FUNCTIONS 1481 THE JACOBIAN Suppose that u ≡ u(x,y) and v ≡ v(x,y) are two functions of two independent variables, x and y;5 Let X 1,X 2,,X be independent Poisson random variables with mean one Use the central limit theorem to approximate P{P i=1 X i > 15} Solution The expectation of each X i is 1, and so is the variance Therefore, E(P i=1 X i) = , and so is the variance If we apply the CLT, then
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12 Locate the stationary points of x 4 y 42x 2 4xy2y 2 13 Find the volume of greatest rectangular parallelepiped that can be inscribed in ellipsoid x 2 a 2 y 2 b 2 z 2 c 2 = 1 14Find the maximum and minimum value of x y ¿ (¿¿) y sin ¿ x sin ¿ sin ¿Y 2 E h1; I need the solution for this If u = 2yz/x, v= 3zx/y, w= 4xy/z Show that ∂ (x,yz)/∂ (u,v,w) = 1/96 Please
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Example Find the general solution to the differential equation xy′ 6y = 3xy4/3 Solution If we divide the above equation by x we get dy dx 6 x y = 3y43 This is a Bernoulli equation with n = 4 3 So, if wemake the substitution v = y−1 3 the equation transforms into dv dx − 1 3 6 x v = − 1 3 3 This simplifies to u=xy ∂u/∂x = y ∂u/∂y = x v=x^2y^2 ∂v/∂x = 2x ∂v/∂y = 2y z=u^2lnv ∂z/∂x =u^2 (1/v)∂v/∂x (lnv) ( 2u) ∂u/∂x =u^2 (1/v)Transcribed image text Let f(x, y) = 3x^2 4xy 1, x = u^2 v^2 and y = 2uv (a) Compute partial differential f/partial differential u and partial differential f/partial differential v (write your answers in terms of u and v) (b) Evaluate partial differential f/partial differential y when (u, v) = (1, 2)
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However we can perform a transformation to remove the constants from the linear numerator and denominator Consider the simultaneous equations {x 2y −3 = 0 2x y −3 = 0 ⇒ {x = 1 y = 1 As a result we perform two linear transformations Let {u = x −1 v = y −1 ⇔ {x = u 1 y = v 1 ⇒ ⎧⎨⎩ dx du = 1 dy dv = 1 And if we@u @y = 2x 6y 2; Cauchy RiemannLinksCauchy Riemann equations in the cartesian form https//youtube/72XKWDKZf2gCauchy's Integral formula https//youtube/IqmSc4yZE0Cauch
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Click here👆to get an answer to your question ️ If u = f(r) , where r^2 = x^2 y^2 then ( ∂^2u∂x^2 ∂^2u∂y^2 ) = Solve Study Textbooks Guides Join / Login Question0 votes 1 answer Find an analytic function whose real part is e^x(xcosy ysiny) asked in Mathematics by Sabhya (712k points)Answer (1 of 2) For the equation y'' 4xy' (4x^2 3)y = xe^(x^2) , it should be noted that the function f(x) = e^(x^2) is so that f'' 4xf' (4x^2)f
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设u及v是解析函数f (z)的实部及虚部,且uv= (xy) (x^24xyy^2)z=xiy,求f (z) xingyuxinyuan237 1年前 已收到1个回答 举报 赞 yahas 果实 共回答了31个问题 采纳率：903% 举报 用ux表示u对x的偏导数,uy、vx、vy类似, 学过柯西黎曼方程吧：ux=vy,uy=vx, 对所给条件分别对x,y求偏倒得F(x) = xy2 ix2y then u(x,y) = xy2 and v(x,y) = x2y The first order partial derivatives are ux = y2, uy = 2xy, vx = 2xy, and vy = x2 Therefore the CauchyRiemann Equations ux = vy and vx = −uy are satisfied only at when x and y are zero, thus when z = 0 b) The function is differentiable only at z = 0, because the first orderpartial deriva 解析函数的导函数怎么求 来源：网友 推荐 更新： 解析函数的导数公式 —— 答案 f (z)=u (x,y)iv (x,y)为解析函数,则f' (z)=u'xiv'x=v'y (1/i)u'y 求解析函数的导数,如图 —— u= (x^24xyy^2)2x2yv (x,y)du/dx=dv/dy du/dy=dv/dxu=x^22xy2xv=y^22xy2yf (z)=x^22xy2xi (y^22xy
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Best answer First we have to solve the homogeneous equation (31) x2y" − 4xy' 6y = 0 This is an EulerCauchy equation, so we look for solutions of the form y = xm The characteristic equation for m is m (m − 1) − 4m 6 = 0 Note m (m − 1) − 4m 6 = m2 − 5m 6 = (m − 2) (m − 3), so the roots are m = 2 and m = 3V(x,y) = x2 y2 is positive definite on all R2 Function V(x,y) = x2 y2 −y3 is positive definite only in a small strip along the xaxis Function V(x,y) = xy2 is no positive definite in any open U containing the origin Example 2 A flexible family of positive definite functions is given by V(x,y) = ax2 2bxy cy2,B and x are vectors of dimension n × 1 The minimum value of f (x) will occur when x equals
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Definition 161 Suppose H R2 → R has continuous second partial deriva tives on a domain D We say H is harmonic in D if for all (x,y) ∈ D, H xx(x,y)H yy(x,y) = 0 Harmonic functions arise frequently in applications, such as in the studyU @(x;y) @(u;v) dvdu = ZZ S u 1 2v dvdu = 1 2 Z 3 1 Z 3 1 u v dvdu = 1 2 Z 3 1 ulnv3 1 du = 1 2 Z 3 1 u(ln3 ln1)du = 1 2 Z 3 1 uln3du = 1 2 ln3 u2 2 3 1 = 1 2 ln3 1 2 (9 1) = 2ln3 Remark 22 Had we not used the trick \Jacobian of inverse equals inverse of Jacobian", we would have had to nd the inverse transformation explicitly uv= y2,y= pSolution for 22 (x²y² sxy 2) ydx ( x ² 4xy 2)xdy=0 Q 1=₁xe³dx Obtain the numerical integral of the integral by dividing the integral into 5 equal parts A Given I=∫36x·e3xdxWe have to evaluate the integral using the Trapezoidal method and Simpson's
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The iteration step in order to solve for the cube roots of a given number Nusing the Newton Raphson's method is A scalarvalued function is defined as f (x) = xTAx bTx c , where A is a symmetric positive definite matrix with dimension n × n ;Learn how to solve implicit differentiation problems step by step online Find the implicit derivative (d/dx) (x^24xyy^2=4) Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable The derivative of1;1i, ~r u 1 1 0~r v= ~ ~{ ~ k 1 1 1 = h1;
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Solve your math problems using our free math solver with stepbystep solutions Our math solver supports basic math, prealgebra, algebra, trigonometry, calculus and moreCompute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals For math, science, nutrition, history84 Sanyasiraju V S S Yedida sryedida@iitmacin 72 Classify the following Second Order PDE 1 y2u xx −2xyu xy x2u yy = y2 x u x x 2 y u y A = y 2,B= −2xy,C = x2 ⇒ B − 4AC =4x2y2 − 4x2y2 =0 Therefore, the given equation is Parabolic
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Answer The equation is M(x,y)dx N(x,y)dy =0 with M = 4xy y^2 , M_y = 4x 2y N = 2y 2x^2 , N_x = 4x # M_y The equation is not exact , but (N_x M_y)/M = 2/y , depends only on y The integrating factor is 1/y^2 and leads to the equation P(x,y)dx Q(x,y)dy =0 , with P = 4x/y 1
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