Apply the Laplace operator to the function h(x, y, z) = e3z sin(4y). V²h

Q: Show that f(x, y) = e«" (A cos(ay) + B sin(ay)) satisfies Laplace's equation: = 0.

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Q: V. Consider the vector-valued function t² + At – 5 Int Vt, t #1 t2 – 1 Ř(t) = î + ŷ– k, 3i + t = 1.…

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Q: Find the Laplace of f(t) = (t− 2)(t+2)e-³t

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Q: þ/ Show that the function satisfy Laplace equation f(x,y) = In /9xy+3y²

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Q: Solve using Laplace Transforms y' + y = tsin(t); y(0)=0 %3D

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Q: Evaluate the line integral Jc F · dr, where F(x, y, z) = xi – yj + 3zk and C is given by the vector…

A: Given F(x,y,z)=xi-yj+3zk and C is the curve given by  rt=sint,cost,t. Find ∫CF·dr.

Q: Use the Chain Rule to find dz/ds and dz/dt. z = x°y°, x = s cos(t), y = s sin(t) dz as az at %3D

A: Given:-z=x5y9x=scos⁡t,y=ssin⁡tTo find∂z∂s&∂z∂tBy chain rule∂z∂S=∂z∂x∂x∂s+∂z∂y∂y∂Swe…

Q: Solve by Laplace transforms. y – 2y = f(t), y(0) = 0.

A: The given Laplace transform is: y'-2y=ft,   y0=0 The given function is: ft=10    0≤t<28…

Q: find inverse laplace by convolution integral: F(s)= 1/((s+2)(s-3)2 )

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Q: Find the differential of w = x ev sin z at the point (x, y, z) = (1, 1, 0). %3D O dy + dz O dy O dx…

A: Obtain the differential of the given function as follows. w=xeysin zdw=ddxxeysin zdx+ddyxeysin…

Q: Which transfer function describes an integration in the Laplace domain? | F(s) = 1 | F(s) = 1/(1 +…

A: Solution

Q: Use the Chain Rule to find aw at w = xycos z; x = t6, y = t³, z = arccost

A: The given functions are w=xycos z ; x=t6 , y= t3 , z= cos-1t. To Evaluate: ∂w∂t. As per policy, we…

Q: Evaluate the line integral f. 3x?yz ds along C:x = t,y = t²,z = ÷t³ , 0 <t<1

A: Use the following formulae, to obtain the solution.…

Q: Find the total differential of the function f(x, y, z) = 4 ryz2 df ? dx + ? dy + ? dz

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Q: Show that the function z = e* sin y satisfies Laplace's equation a²z a?z = 0 дх? ду?

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Q: Find the Laplace of f(t) = (e-3t - e5t )2

A: To Find L(e-3t-e5t)2: Expand (e-3t - e5t )2: (e-3t-e5t)2=e-3t2-2(e-3t·e5t)+e5t2…

Q: Solve this Question by using Laplace y" + 4y' + 5y = 125t2 y(0) = 0,y'(0) = 0 %3D

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Q: Find Laplace treins form for Ý +4Y+3Y = e y(o) =y co) = 1 %3D %3D

A: Formula Used:L{y''}=s2Y(s)-sy(0)-y'(0)L{y'}=sY(s)-y(0)L{eat}=1s-aL{tneat}=n!(s-a)n+1

Q: dz Find dt where z(x, y) = x² – y°, with z(t) = 9 sin(t) and y(t) = 3 cos(t). %3D %3D %3D dz dt

A: Given,

Q: Show that the function f(x, y, z) = x² + y? – 2z² satisfies the Laplace equation.

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Q: Find the flow of the velocity field F = (y + z)i + xj - yk, where velocity is measured in meters per…

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Q: Integrate f(r, y, z) = for 0 <t< V3. + y² over the curve r(t) = [cos(t) +t sin(t), sin(t) – t…

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Q: TIJY sin /x dx dy by using the transformation u= Vy. v= Vx ? 0 0

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Q: S how that f (x, y)= tan! Satio fy the Laplace equation fae + fyy =0.

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Q: F'ind the differential dy. (a) y = xv8 – x %3D dy =

A: we have to find

Q: a. Express the function f(t)=- its Laplace transforms. π-t, sin t, 0 π 15 terms of unit step…

A: Introduction: The unit step function is a significant special function. It is a piece wise constant…

Q: Find dw/dt if w = xy + z, x = cos t, y = sin t, z = t.

A: Given: w = xy+z,  x = cost,  y = sint,   z = t To find: dwdt

Q: Show that f(x, y) = ea« (A cos(ay) + B sin(ay)) satisfies Laplace's equation: f, Pf = 0. dy?

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Q: Solve using Laplace transforms. y" – y' + 5y = 58(t – "/6) sin t y(0) = y'(0) = 0 - %3D

A: The Laplace transform is an integral transform that converts a function of a real variable t to a…

Q: 3 find the invers laplace f(t) = S+3 s²+3

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Q: Evaluate the line integral r(t) = (sin t, cos t, t), F · dr, where F(x, y, z) = −3xi + 2yj + 4zk and…

A: Given:- Fx,y,z=-3xi+2yj+4zk rt=sint,cost,t To find:- ∫cF.dr , where c is given vector rt ,0≤t≤3π2

Q: What is the laplace of L(f"(t))? A s?F (s) – sf (0) – f (0) B sF (s) – f (0) s²F(s) – sf (0) – f…

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Q: Find the work done by force F(x, y) = 2yi + 3xj + (x + y)k in moving an object along curve r(t) =…

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Q: S(x, y, z) = 2x'y²z²

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Q: Q2) A) Verify that wry Wyx for w x sin y +y sin x + xy. s) %3D B) Show that f(x, y) = e-y cos 2x…

A: We have to verify that the equations satisfy the given conditions.

Q: Show that the function z = ex sin y satisfies Laplace’s equation                      ∂2z / ∂x2 +…

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Q: Evaluate the line integral along the curve C. y² ds, C is the curve r(t) = | 9 sin x + z2 i+ + 5tk,…

A: To find the line integral along with curve C, use the formula,…

Q: Evaluate the line integral along the curve C. x² +y² ds, C is the curve r(t) = (2 sin 6t)i + (2 cos…

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Q: Compute the line integral of the scalar function f(x, y, z) = 2x² + 8z ver the curve c(t) = (et, t2…

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Q: Show that f(x, y) = ea# (A cos(ay) + B sin(ay)) satisfies Laplace's equation: + - 0. dy?

A: Given,            f(x,y) = eαx(Acos(αy) + Bsin(αy))             Then show that  ∂2f∂x2+∂2f∂y2=0

Q: f (x, y) = –3x²y + y³ satisfies Laplace's equation fex + fyy = 0

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Q: Verify that the function f(r, y) = In (x² + y²) is a solution of the Laplace equation %3D Zzz+żyy =…

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Q: Hnew ³ Let w=f(u) + 9 (v); where u = x+iy and V=X-iy, show that w satisfies the Laplace eap. Wxx +…

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Q: (c) Show that the function z = e* sin y + e cos x satisfies Laplace equation azz azz = 0. ax? ay?

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Q: 2xy Show that f (x, y) = tan satisfies the Laplace x² – y? Equation f + f, =0 XX yy

A: We have fx,y=tan-12xyx2-y2 We need to prove that fxx+fyy=0 Now,  fxx means derivative of f with…

Q: Calculate Sf(F(t)) | (t)) dt f √ √ √x + 1 (x+y+z) ds where C is the helix r(t) = (cos(t), sin(t),…

A: What is Line Integral: A line integral in mathematics is an integral in which the function to be…

Q: Show that the function u(x, y, z) = e3x+4ysin(5z) satisfies the Laplace equation in R3

A: A function u(x,y,z) is said to satisfy the Laplace equation in ℝ3 if u(x,y,z) satisfies…

Q: The inverse Laplace of F(s) : (s-2)2+2 II

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0). The multivariable limit of this function as it approaches the points (x0, y0) and is denoted as: