Consider the following functions: e e f1(x) = sin(x1) cos(x2), æ € R? f2(x, y) = x' Y , æ, y E R" %3D f3(x) = xx", x E R" Exercises a. What are the dimensions of Ofi ? b. Compute the Jacobians. dx

Q: Q1// B: If u+v=e cos y and u-v=e'sin yfind the Jacobain of the a u, v at (x.y) (1,0) dx,y function

A:

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

A:

Q: Suppose that over a certain region of space the electrical potential V is given by the following…

A: Given data:     The expression for the given electrical potential function is: V(x,y,z)=3x2-2xy+xyz.…

Q: Suppose that r(t) = cos³ t i+ sin³ tj+et k, show that d dr (r.- dt dt d?r X at2 dr d'r. r.at at3…

A: Given: rt=cos3t i+sin3t j+e-t k we have to show that ddtr·drdt×d2rdt2=r·drdt×d3rdt3

Q: function u (t) == C, cos(@ot – ao) that would result if the mass on the spring were set in motion…

A:

Q: I am having trouble evaluating ∭x2 dV, where E lies between the spheres ρ = 1 and ρ = 3 and above…

A:

Q: b) Find the potential P(x,y) attached to the exact ODE 2r sin(y) + x* cos (y) y = 0

A:

Q: Let z = e* sin(y+a), show that %3D ôx? oy?

A: Given, z=e-xsiny+a, then we have to show that ∂2z∂x2+∂2z∂y2=0.

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ € R? f2(x, y) = x'y , æ, y E R" f3(x) =…

A:

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A: Here is the solution of the given problem.  To solve this problem we use simple concept of…

Q: Differentiate d/dx  ∫ [cos(x), 0] t dt (using the Fundamental Theorem)

A: Differentiate d/dx  ∫ [cos(x), 0] t dt (using the Fundamental Theorem)

Q: Consider the functions z = 4 e* In y, x= In (u cos v), and y =u sin v. dz and du dz (a) Express as…

A: Given, z=4exlny,  x=lnucosv,  y=usinv Part (a): z=4exlny,  x=lnucosv,…

Q: Q2) A) Verify that wxy wyx for w x sin y+y sin x +xy. %3D R) Show that f(x v) = e-2y cos 2r…

A: See the details solution in below

Q: Find the Lyplace tiangern f (t) = sin Ct+a) of cor (t-a) usng S e-e f (t) dt f Ct) dt .

A: The given problem is to find the Laplace transform of the given function using integral method.

Q: y = In(sec t), x = log(cosec t) In(sect), x %3D %3D

A:

Q: Select all the functions that solution of y"-y%=D0 are D A. y= e O Biy = e A C, y = x O D. y = COS…

A:

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ € R? %3D T. f2(x, y) = x' y, x, y E R"…

A:

Q: Consider the following. w = cos(x – y), x = t?, y = 97 (a) Find dw/dt by using the appropriate Chain…

A:

Q: tan(): '(), Suppose z = x = u cos(5v) , and y = u sin(5v) . Compute dz/du and az/dv in two different…

A:

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

A:

Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

A:

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ € R² f2(x, y) = x'y, x, y E R" f3(x) =…

A:

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ e R? f2(x, y) = x' y , x, y E R" f3(x)…

A:

Q: .: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G (x, y) = x³ y² + y² sin…

A:

Q: Show that the process X(t) = A cos At + B sin At (where A and B are RVs) is wide-sense stationary,…

A:

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: Show that the function f(x, y) = e-2y cos 2.r satisfies the Laplace equation.

A: use partial derivative to check whether it  satisfy the equation or not

Q: Consider the 2nd order equation 2²y+ Sin dy y² +y = Choose the ight assertion A) the linearized…

A: consider the 2nd order equation d2ydx2+sindydt+y2+y=0 this is a non linear homogenous differential…

Q: Evaluate dW at t = dt for the function W = xy x = sin t, y = cos t, z = t². 4 49л2 972 9x2 4 49л

A: Given: W=xyz,x=sin⁡t,y=cos⁡t,z=t2 Substitute the value of x, y and z in w , we get W=sintcostt2…

Q: Determine if the following functions are linearly dependent or independent by determining their…

A: The Wronskian of a set of function y1, y2, y3 is defined as, If the Wronskian W is non-zero then…

Q: = prove that the following examples are satisfy Laplace equation , y) =(e-2y)Cos2x 2-f(x, y) = Ln…

A:

Q: Consider the following. w = cos(x - y), x = t2, y = 1 (a) Find dw/dt by using the appropriate Chain…

A:

Q: Determine DFT X[ej"] of the signal (answer a or b) a) x[n]coswon. b) x(n) = a"u(-n- 1) %3D

A: As it's mentioned in the question to answer between a) or b), so showing steps for only one part. b)…

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

A:

Q: (c) Show that the function z = e* sin y + ev cos x satisfies Laplace equation ax? a2z = 0.

A:

Q: Find the antiderivative that satisfies the initial given condition. f"(x) = 2e" + 4 sin x, f(0) = 0…

A:

Q: Given F(x, y) = (sin y + e*, x cos y) and C: x=t, y=t(3 —t), 0≤t≤ 3. (a) Show that F is conservative…

A: Given : F(x,y)=sin y+ex,xcosy C:x=t, y=t(3-t), 0≤t≤3. To show:  (a) F is conservative and find a…

Q: Let f (x,y) = sin(xy) + x² ln(y). fyx at (0,) is equal to: %3D А. 33 В. О С. 3 D. 1

A:

Q: Find the antiderivative that satisfies the initial given condition. " f"(x) = 2e" + 4 sin z, f(0) =…

A:

Q: Consider the following. w = xy cos z, x = t₁ y=t5, z = arccos t (a) Find dw/dt by using the…

A:

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

A:

Q: Q12:- Show that the following functions satisfy the Laplace's equation+ 1) z= Ln(x + y) +2 tan-12 2)…

A: Show that the given function satisfies Laplace's equation. (i) z= Ln(x2 + y2 ) + 2 tan-1 (y/x) (ii)…

Q: Calculate dz/du and dz/dv given the following functions: 2x-y x + 3y' = f(x, y) = = Z = x(u, v) =…

A:

Q: Let w(x, y, z) = x2 + y? + z2 where x = sin( – 2t) , y = cos( – 8t), z = e". 9t dw dx dy dz…

A: To find the derivative using chain rule

Q: r(t) = cos(nt)i – In(t)j+ vt – 16k; to = 17 | -

A: Given :

Q: Consider the following functions: f1(x) = sin(x1) cos(x2), æ € R? f2(x, y) = x 'y, x, y E R" f3(x) x…

A:

Q: Given that F(x, y, z)=sin x i+cos yj+ xzk. Evaluate , F.dr where C is given by r(t)={r',-f,t) :…

A:

Q: Solve exactly 1. u(x) + f or approximately the following four exercises using Schmidt's method.…

A:

Q: 6. Let r = t sin t, y = t cos t, 0<t<1. dy (a) Find dr

A:

Q: The function f(z)=e² is: „(x.y) =e*sin(y) O contiuous in CO u(x.y) =e*cos(y) O discontiuous in CO

A: Use the property of complex function to break in real and imaginary part. Here also use the concept…