Suppose F = ∇ * A, where A = (y + sqrt(z))i + e^(xyz) j + cos(xz) k. Determine the flux of F outward through the hemisphere x2 + y2 + z2 = 1, z>=0.

Q: Calculate the total differential dz for z = r sinh(ry) + yelv

A: Note- since you have asked multiple questions, So Acc. to Bartleby guidelines we are supposed to do…

Q: Show that [ xdhy – ydy c Vx*+y² is independent of any path of integration which does not pass…

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Q: Show that the differential form in the integral is exact. Then evaluate the integral. (1,2,3) Jazy…

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Q: The force exerted by our home planet is GMx if x R, measured on a unit mass at a distance x from the…

A: We are given the function h(x) as the following. Notice that GMxR3 is continuous everywhere. And…

Q: 2. Given that f(t) = f cos x cos(t − x) dx, fin integral.

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Q: in integating the function fxcos(3r) de using integration by parts, what would you choose for your…

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Q: 2. When object P, with mass m, , moves at v, and object Q, with mass m,, moves at v,, the velocity…

A: Given,For P, mass = m1  and velocity = v1Let, velocity after collision is u1For Q,mass = m2 and…

Q: (b)Find the derivative of g in the direction of a= 2i+ curl w where at point (0,-1,1) g = In- 4 + 2…

A: Given, w=cosxyzi+xeyz+1j+xyzk Then, Curl…

Q: Let C′[a, b] be the set of all functions whose derivatives are continuous on [a, b]. Show that the…

A: Use the operator notation. where f is in C'[a,b]

Q: Show that if the function f(x, y) = ex siny + ecosx, satisfies Laplace Equation.

A: We can solve this as follows:

Q: (a) Evaluate the indefinite integral (anti-derivative) d.x Vr2 +1 by using the substitution u = x?…

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Q: Use symmetry to evaluate the double integral. Ja+ (1 + x8 sin(y) + y sin(x)) dA, R= [-π, π] x [-π,…

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Q: dz Z 1. around unit circle of radius 1 and center at (0, 0) counterclockwise. 2. By using direct…

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Q: 10. What is the work done moving a particle from x = 0 to x = 1 m if the force acting on it is F =…

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Q: Let f(z) = z2 is f(z) satisfy Cauchy-Riemann equation ?

A: Let any complex function f(z) = u + i v then it satisfies Cauchy-Riemann equation when 1) ∂u∂x=∂v∂y…

Q: Find i Pontial dlexivalive Sur face level Ü) Li) F Gradient of iy Tongen planes 4 funct ion Z= Cos…

A: (i) Given the function is z=cos-1x+y. Find the partial derivatives.…

Q: If F= represents the flow of a liquid through a parabolic cylinder S: y=x, 0sxs2, 0szsl. That is…

A: If F→=y2,-x2,z2 represents the flow of a liquid through a parabolic cylinder S:y=x2,0≤x≤2,0≤z≤1.…

Q: Find the total differential of W = x^4 y^3 z^2 + x^3 Y^2 + x^2 z +y + 3 at (2,1,1)

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Q: dv Integrate ду 1 partially with respect to y. II

A: To partially integrate the given function with respect to y, implies x will be treated as a constant

Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

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Q: Consider a rod of length meters with density 8(x)=(2- cos x) kg/m over the interval 0<x<5.

A: Given that The length of rod is π2 and the density of the rod is δ(x)=2-cos(x)  with 0≤x≤π2 To find…

Q: Let X, Y be jointly Gaussian, standard, and Cov (X,Y) = y. Find pdf of Z = X +

A: Answer:-  Given that, X and Y have jointly Gaussian distribution.

Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Suppose that a duck is swimr - perature is given by the formı v²sin x. Find dT/dt in two chain rule…

A: Given query is to find the derivative of function T.

Q: The forces of the human gastrocnemius tendon during traction and during recoil can be approximated…

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Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: 4. Curl of f (x, Y, z) = 2xyi + (x² + z²)j + 2zyk is

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Q: Show that f(x,y)=exyx is differentiable at point (1,0) using the definition of differentiability.

A: A function is called differentiable, if the derivative of function exists at each and every point in…

Q: 2) Show that the integral f. (cosyz dx – xz sinyz dy - xy sinyz dz) is path independent and find its…

A: Condition for line integral to be path independent is given below. ∫cFdr is path independent if F…

Q: 3. ) Compute for the increment of the function f(x,y) = *+3y . Also, find the total differential of…

A: w = f (x, y) and let ∆x and ∆y be increments of x and y respectively. The increment ∆w of w = f (x,…

Q: Find the potential function for F(x,y,z) = (x^-1z)i + (y^-1z)j + ln(xy)k. Show all steps and explain

A: Potential function of  F, is f such that ∇f=F.

Q: Prove that the homogeneous integral equation

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Q: Evaluate the line integral / (6x + 2y)dx + (x – 7y)dy along the 10 cos t, y in (0 <t: curve C: x =…

A: A detailed solution is given below.

Q: Find the total differential of z= f(x, y), where f(x, y) = In Use * for multiplication of variables,…

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Q: Use a substitution of u = r to integrate : dr.

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Q: Describe any one of the physical system where we can apply Bessel or legendre differential equatin?

A: In the question it is given Legendre and Bessel differential equation. The objective is to…

Q: a) by parametrizing C, b) by using the Fundamental Theorem of Line Integrals.

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Q: Solve by Laplace transform Utt – 9urz = 0, 0 0 %3D u(0, t) = u(r, t) = 0, %| %3D u (x, 0) = 0,…

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Q: Find the derivative of g in the direction of a = 2i+ curl w at point (0 where น u+2 8 = In- J u =…

A: Given: the function g=lnuu+2, u=z(xy+z2)2, w=cosxyzi+xeyz+1j+xyzk To find: the directional…

Q: Determine if F=(e* cos y + yz)i + (xz-e* siny) j+(xy+z)k is conservative. If it is conservative,…

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Q: is function oF x $ y xz + y In x = ze-sin (xy) D. Ausumo that z Find zx iF

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Q: Estimate the error if cos vt is approximated by 1– in the integral J cos vt dt. 2' 4! 6!

A: For the given  Cos t = 1-t2+t24!-t36!+.... error is estimated as shown below

Q: The force exerted by our home planet is GMx if x < R, R3 h(x) = GM if x 2 R, measured on a unit mass…

A: In the question we have to identify if the function is continuous.

Q: 2) Find the total differential dz for z = tanhy In(x)

A: Total differential of z is given by dz=(∂z/∂x)dx+(∂z/∂y)dy

Q: Find a potential function for F(x, y, z) = (y² cos z, 2xy cos z, –ry² sin z). %3D COS

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Q: Find the total differential for the function w = 13z ysin x. dw = 13z ysin xdx-13z sin xdy-26z ysin…

A: Consider the function: Let,  w=13z3ysinx=fx,y,z The total differential of the function is given by…

Q: A circular plate of radius 5 cm is immersed in a liquid of specific weight w such that its…

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Q: . The work required to move a mass m "infinitely far" from a planet of radius R and mass M is given…

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Q: Let F(r, y, z) = (y² + 2z, 2ry + 4, 2a), and let the curve C be described by x = t, y = t, z = t3…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.