The electric potential at each point (x,y) in the region R: x 2 0, y s1 is given by v - 48xy – 32x - 24y² . Locate the maximum and minimum potential in the region R.

Q: Find the value of integral (x+y +=) ds where C is a part of helix parameterized by r(t) = (cost, sin…

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Q: Y – 5x |dA 100 cos x + y

A: We have to evaluate the double integral ∬100cosy-5xx+ydxdy over the region D where the region D is a…

Q: Evaluate the line integral -y dx + z dy + 2x dz, where C is the helix by 7(t) = (cos t, sin t, t),…

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Q: Evaluate the line integral , F. dr where F = (-4 sin x, –3 cos y, 10xz) and C i: the path given by…

A: Given that F→=-4sinx, -3cosy, 10xz and C is the path given by r→t=2t3, t2, -2t for 0≤t≤1 We have to…

Q: panticle of mass m. moves in the one-dimersianal potential U(x) = U. %3D 92 Sketch Ulx).

A: The function is a rule that associates or fixes a number to the particular point x. If , then the…

Q: find the area of the surface S: x² + y² + 2² = 4 -A <tan -^ (*) FO 6 < COS Cos FJ VI (2) 4 DEX TI 3…

A: The given Surface is                    x2+y2+z2=4 ........1where π6≤tan-1yx≤π3 and π4≤cos-1z2≤3π4…

Q: Exercise 10.6 Evaluate the line integral using a method of your choice. e¯*(cos y dx + sin y dy),…

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Q: - dA where R= {«x, y)|0 sx54 and 0 S y 5 Va} 1+x y Evaluate the double integral [[ R

A: Solve Using multiple integral

Q: Consider the pendulum x' Y, y' = -vy - sinx, - where 0 is a friction parameter and (x, y) E S¹ x R…

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Q: Calculate the double integral || (x + y)?ex -ý dx dy where R is the square with vertices (2, 0), (0,…

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Q: Calculate the line integral of h (x, y)= (2n) i+(x - 2y) j; over r (1) = cos u i+ sin u j, u e…

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Q: The line integral g-dr where g= (2ry - y³)i + (a² – 2ry)j and r(t) = cos ti + sin tj,0<t<a has a…

A: Use green theorem.

Q: Find the potential u(x, y) for u +u =0 in the rectangle 0<x< 2 and 0 < y s 2, if the upper side is…

A: The given problem is to solve the potential u(x,y) Laplace partial differential equation, we have to…

Q: Calculate the double integral. JS y dA, 4xye+²y R R= [0, 1] × [0, 8]

A: Given; ∫∫R4xyex2ydA, R=0,1×0,8   As we know the common integral. ∫eax+bdx=1aeax+b+C  And ∬dA=∬dxdy…

Q: Find the potential function of the field F (x, y, z) = i+j+k. Assume that the constant of…

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Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

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: Evaluate the line integral along the given path. I ( 2+ y2 + z² ) ds C: r(t) = sin(t)i + cos(t)j +…

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Q: Given the path, dx + xеУdy y determine the line integral using Green's theorem if C, is y = x² + 2,…

A: Given path is ∮Cxydx+xeydy To determine the line integral using Green's theorem if C1 is y=x2+2,…

Q: Given F = (2xze*²z – 4, cos z, x²e*z – 3ycos?z sin z) 1. You are to show that the line integral S.…

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Q: Evaluate the line integral along the given paths.       C(x2 + y2) ds   a) C: line segment from (0,…

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Q: Calculate the double integral. ∫∫ x sec2y dA, R = {(x,y)| 0≤ x ≤2, 0≤ y ≤ π/4 }

A: Given: The given function is  f(x,y)=x sec2(y) and the region is R=(x,y)|0≤x≤2 , 0≤y≤π4  To find: we…

Q: A uniform electric field points in the :-direction, E = Ek. By explicitly calculating the surface…

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Q: Evaluate the triple integral 1 V1-y V4-z? y? COS cos (z? + y? + z?) dzdzdy. V12 + y? + z2

A: Given triple integral is ∫-11∫01-yz∫04-x2-y2cosx2+y2+z2x2+y2+z2dzdxdy To evaluate the integral.

Q: Calculate the integral I = | (2a – y) dx - (a – y) dy where L is the curve represented as x = a(t –…

A: (2a - y) = 2a - a(1 - cost) = a(1 + cost) (a - y) = a - a(1 - cost) = acost dx = a(1 - cost)dt dy =…

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: 5. * m/2 X sen y dy dx n/2 6. Ja/6 EL cos y dy dx

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Q: 21. Find the line integral of f(x, y) = ye along the curve r(t) = 4ti – 3tj, –1 < t < 2.

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Q: Find the real number a for which the value of the iterated integral V1- (y- (-2²- ỷ +2y) dx dy a-1 -…

A: Let fx,y=-x2-y2+2y. To determine the maximum value of the function, differentiate f(x,y) partially…

Q: Let 0 < c < 1. Compute the integral rc+ioo dz Jc-ioo sin nz' where the integration path is the…

A: We will use the basic knowledge of complex line integrals to answer this question properly, in…

Q: Calculate the double integral f SR T cos(2a + y) dA where R is the region: 0 <I<0 <y

A: To find- Calculate the double integral ∬Rx cos2x + y dA where R is the region: 0 ≤ x ≤ π6, 0 ≤ y ≤…

Q: Use the separated solutions to solve the Laplace equation. in the region 0 0 given the boundary…

A: The Laplace Equation Uxx+Uyy=0with(a) U(0,y)=U(1,y)=0(b) U=0 as y tends ∞c) U(x,0)=sin5(πx)

Q: Q2/ Find the value of the line integral . F.dr when F = zi + xj + yk and C is the helix in figure…

A: Given:F→=zi+xj+jkr(θ)=cosθi+sinθj+3θk

Q: Find the work done in moving a point charge Q- - 20 1C from.the origin to (4,0,0) m in the ficld E-+…

A: To find - Find the work done in moving a point charge Q = -20 μC from the origin to 4, 0, 0m in the…

Q: Find the mass and center of mass of a wire of constant density d that lies along the helix r(t) = (2…

A: Given information:

Q: V = 200 /R volts in free space in the region bounded by 2mm <R <3mm, 0 <8 <n / 2 and 0 <o <n / 2…

A: Given 2 mm&lt;R&lt;3 mm, 0&lt;θ&lt;π2 and 0&lt;ϕ&lt;π2 Now, 2×10−3 m&lt;R&lt;3×10−3 m And under…

Q: Find the line integral of f(x,y) = y e* along the curve r(t) = - 5t i – 12t j, - 1sts0. The integral…

A: Use parametric to integration

Q: Find the value of the closed loop integral z3 + sin z -dz (z – i)3 taken around the boundary of the…

A: GIven  ∮z3+sinz(z-i)3dz   To compute this integral we have to use the following theorem. Theorem:…

Q: 14. Find the line integral of f(x, y, z) = V3/(x² + y² + z?) over the curve r(t) = ti + tj + tk, 1 <…

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Q: Evaluate the line integral INTEGRAL C (x − y 2 )ds where C is given by r(t) = (t^2 + 1)i + (t +…

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Q: Find the line integral of ƒ(x, y) = sqrt(y)/x along the curve r(t) = t3i + t4j, 1/2&lt;=t&lt;= 1.

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Q: 3 2 Use a line integral to calculate the length of the helix ř(t) = (a cost, a sint, bt) 0<t < 2n…

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Q: *Q. ( 12 ), p. 410 : Find the area of the surface generated by revolving eУ + е-У the curve x = 0…

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Q: Calculate the double integral. sec) da, R={cx, v) 1 0 s*s2,0s ys} sec2(y) dA, R = = {(x, y) | 0 < x…

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Q: Evaluate the triple integral. {x, y, z) | 0 s y s 2, 0 sxs 4- y², 0 sz s Sy 6x dV, where E =

A: From the given problem:   ∫∫∫E6xdV   And region:   0≤y≤20≤x≤4-y20≤z≤5y   So, we can write as:…

Q: Let F = (x + 2xy,x² – y). a. Given that F is conservative, find the potential function ø where F =…

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Q: Find the total differential of w ze(z+y) at (0,0, 1).

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Q: Find the line integral of F(x, y, z) = x'i + yj+ zk along the line segment C from the origin to the…

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Q: Find the absolute maximum and minimum Of the function f(x, y) = x + Y on the elliptic region {(x, y)…

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