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) : 0

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Q: (x,y) = xy i + x+15 yj, on R'. Find a potential function for

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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: Q7 Show that CX,y) =e is Condinuous at co,0)

A: The given function is:  fx,y=e-x2+y2 The given point is 0,0

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Q: Use the total differential dz to approximate the change in z= 6-x² – y² as (x, y)moves from the…

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

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Q: 19. Find. thedivergen ce. and.cur..4the fellowiny vecter. field: a.. F = Xej+. y.e... b. Ês ĕsiny i…

A: We need to find the divergence and curl of the following vector fields. a. F→=xeyj+yezk=(0,xey,yez)…

Q: Let f(2) = z e 2 #0. z # What type of singularity does f(2) have at z 0? (Justify.) What is the…

A: The singularity of the function f (z) have at z = 0  .     Find the Taylor series expansion of z5…

Q: 5. Calculate the double integral f, sin(2x) cos(3y) dA , R = [0, 1/2] x [0, 1]

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Q: 3. For all a? + y + 2? > 0, verify that the velocity potential 1 $(r, y, z) = Vr? + y² + z2 is a…

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Q: Determine the solution of the differential de +y cot x = cosx O a. ycos I : sin'z +e o b. ycos x =…

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Q: Use Green's Theorem to evaluate the line integral. ex cos(2y) dx – 2ex sin(2y) dy C: x2 + y2 = a²

A: Use Green's theorem

Q: ƏF Find at (x, y) = (T, 1) if f (x,y) = cos vt dt. cos yt dt. ww

A: given,          fx,y=∫0x2ycostdt

Q: If f(z, y) = 22e cos z+9 cot y, then the total differential of f at (0, ) (0. is: O df = 22 dz + 18…

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Q: Find a potential function for F or determine that F is not conservative. F = (cos(z), 16y, -x…

A: The vector field given in the question is: F = <cos(z), 16y, -xsin(z)> Hence, F1 = cos(z), F2…

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: 3. If w = e3x-2y cos 4z where x = Int , y = In (t² + 1), z = t3, aw at t = 1? at what is the value…

A: Consider the following equation: w=e3x-2ycos4z Substitute x=lnt, y=lnt2+1 and z=t3 in the above…

Q: find a potential function ƒ for the field F. F = ( y sin z)i + (x sin z)j + (xy cos z)k

A: Concept: A definition of a function is that a relation from a set of inputs to a set of possible…

Q: If f(x, y) = 17e* cos x + 6 cot y, then the total differential of ƒ at ( 0, ÷) i : O df = 17 dx + 12…

A: f(x,y)=17e-xcos x +6 cot y

Q: (B) Find the potential function V = V(r, y) such that %3D + yi+ + r)j = V¥ 1+ x2 + y? 1+ x? + y? and…

A: We have to find the potential function ψ=ψx,y such that x1+x2+y2+yi+y1+x2+y2+xj=∇ψ and also evaluate…

Q: Show that F. dr is independent of its path, given that F(л, у, 2) — + tan 1+x² 2) i+ tanrj+ k. 1+2²…

A: We will show the required result.

Q: Find the work done by F(x, y) = (x, Y) over the curve r(t) = (e' cost, e' sin t) (x² + y?)3/2 from…

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Q: Evaluate the line integral f (-y sin(xy)i – x sin(xy)j) dr using the potential function f(x, y) =…

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Q: What is the value of the line integral on the field 5xy? 5æ?y F(x, 3) = ()i+ ( 2 along any path from…

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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: Use a local linear approximation of the function f(z) = VI cos[(V2r)] at some approximate v3.98…

A: To use local linear approximation of the below function to approximate 3.98cos7.963π. fx=xcos2x3π

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: Use Green's Theorem to evaluate the line integral below. dx + 5xy dy C: r = 1 + cos(0), 0 s0 s 2n

A: Consider the provided line integral ∫Cx2-y2dx+5xydy. Let M=x2-y2, and N=5xy. That implies,…

Q: Use the direct method to evaluate the path integral for the work done by F(x, y) = (y, -x) along…

A: -4π

Q: Show that the integral is independent of the path, and use the Fundamental Theorem of Line Integrals…

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Q: Find y', where t (a) y = Jo dt VE2 +1 (b) y /t² +t+1 dt 1 • sin²3x (c) y = | 1 dt t2 +1 Cos x

A: The given function is: y=∫cos xsin2(3x) 1t2+1 dt

Q: 3. Verify the result shown using 'integration by parts'. | t sin(nt) dt = – -t 1 t cos(nt) + 1…

A: Integral by parts

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: Consider F(x, y) = (y², 2xy + 1) . Let c be the curve defined by F(t) = (t³, t), for 0 <t < 2.…

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Q: Use Green's Theorem to evaluate the line integral. et cos(2y) dx 2ex sin(2y) dy C: x2 + y2 = a?

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Q: If f(1, y) = 17e cos z+ 9 cot y, then the total differential of f at (0,) is O df = -17 dr + 18 dy O…

A: Given that:-     fx,y=17e-xcosx+9coty  we know that total differential is =     df=fxdx+fydy…

Q: Q2/ Consider the Fredholm integral equation of the second kind, 1 y(x) = x – | xt y(t)dt by using…

A: Given: The Fredholm integral equation of second kind is, yx=x-∫-11xtytdt.

Q: Find the potential function for F(r, y) =. Select one: O A. f(r,y) = ev + sin y- c O B. f(r,y) = e²y…

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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: If f(r, y) = 22e cos z +2 cot y, then the total differential of f at (0, ) %3D is: O df = 22 da +4…

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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: If f(x, y) = 16e * cos x + 4 cot y, then the total differential of f at (0, is: O df = 16 dæ – 8 dy…

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Q: If f(x, y) = 22e¯ª cos x + 4 cot y, then the (0, is total differential of f at - O df = 22 dæ + 8 dy…

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Q: Evaluate the following line integral I=[3x°dx + 2yzdy +y°dz from A: (0, 0, 0) to B: (2, 4, 8) a) By…

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Q: Use the Chain Rule to find dw/dt. w = In V x2 + y2 + z2 x = 9 sin(t), y = 3 cos(t), z = 5 tan(t) dw…

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Q: does not exist. (b) If u = x cos y+ye", where x=s+2r and y = st , prove that ди ди dt = (s – 21) (x²…

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Q: Find the total differential. w = e7Y cos(4x) + z7 dw

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