Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function for it.

Q: Find the gradient field F = ∇φ for the potential function φ(x, y) = x + 2y .

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Q: Find the gradient field F = Vo for the potential function o below. P(x,y,z) = In (x2 + 2y2 + z²)…

A: We have to find gradient of given function.

Q: 5) Consider the conservative force field F(x, y) = (6x²y? + cos(x), 4x y). a) Give a general…

A: For a vector field F=P,Q that is open and simply connected in a region with continuous first order…

Q: Show that work performed by a constant force field F over any path C ~ from P to Q is equal to F •…

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Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: (2) Let F(x, y) = (cos(y)+2x, -x sin y+1). Show that F is path independent by finding %3D a…

A: For the given vector function to be independent, it must be derived from a potential scalar function…

Q: Show that work performed by a constant force field F over any path C from P to Q is equal to F - PÓ.

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

Q: Find the gradient field F=Vo of the potential function o(x,y.z)= (3x +y +3z) -1/2 Vq = 000

A:

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

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Q: Show whether or not F is a conservative field. If it is, find a potential function for it and use it…

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Q: 2y Find the gradient field F = Vo for the potential function o = F =

A:

Q: 5. Find the potential function for given vector field: F = (e* In(y))i + + sin(2))j + ycos(z)k

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Q: Find the outward flux of the given field across the given cardioid. 5x i+ (e*+5 tan 1ly)j 1 + y2 F =…

A: The following formula from Green's theorem is used to find the outward flux, outwards flux=∮F·n…

Q: The magnetic field along a solenoid is conservative and given by 2xy 7- (2 x ₁-2x, (x²+1)-2). ¡−2x,…

A: Given a vector field as F=2xyx2+1-2x,lnx2+1-2 We need to find the scalar potential from which this…

Q: (a) Suppose a vector field F(x, y) construct a potential function o(x, y). i+ is conservative. Then

A: Consider the given vector field.

Q: Find the Fourier integral of f(x) = e-k* when x>0 and f(-x) = f(x), hence prove that cos(wx) dw w…

A:

Q: The work done in carrying a -1C charge from point A(0,0,0) to the point B(6,0,0) in the field E =…

A: Given,     E=2x.ax^+ 2yay^  VmQ=-1CPoints are A(0, 0, 0)  and  B(6, 0, 0)

Q: 8. Given the vector field G(x, y, z) = e>+2z (1 + x j + 2xk) : a) Show G(x, y, z) is conservative…

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Q: Find the gradient field of the function, g(x,y,z) = e 3z - In (x +2y).

A: Given function  gx,y,z=e3z-lnx2+2y2                     (1) The gradient field is  ∇g=∂g∂x,∂g∂y,∂g∂z…

Q: | 23. Find the outward flux of the field F = ( 3xy i + (e* + tan¯ly)j х 1 + y?, across the cardioid…

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Q: Find a potential function f for the field F. F = (5y + 2z)i + (5x + 3z)j + (2x + 3y)k A general…

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Q: 5) (a) Show that F = (8-14xy2 + 2ye²x) + (e²x - 14x²y) y is conservative. (b) Find a potential…

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Q: Consider the line integral cos(x+2y)dx+2cos(x+2y)dy where S is a curve starting at (0,0) and ending…

A: We have to show that F is conservative. When F is given as F=cosx+2yi+2cosx+2yj For the conservative…

Q: 23. Find the outward flux of the field i + (e + tan'y)j 1 + y², F 3xy %3D across the cardioid r =…

A: Given: The following formula is used to find the outward flux, Comparing with the general form,…

Q: Find the gradient field F = Vo for the potential function o = x²y-3y²x. F =

A: Given the potential function:φ=x2y-3y2xWe need to find the gradient field F=∇φ

Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

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Q: Find a potential function f for the field F = ev *2=(4 i + 4x j + 8x k). A general expression for…

A: We can calculate the following question by usual method of calculating potential function

Q: 11. Find the flux of the field F =ri – yj across the curve C: r(t) = (cos t)i+ (2 sin t)j, 0 <t< 2n

A: Given vector field is F=xi-yj across the curve C:rt=costi+2sintj , 0≤t≤2π Flux of the vector field F…

Q: the potential function f for the field F = 5x i + 9y j + 4z k. eral expression for the infinitely…

A:

Q: The slope field oY = f(t,y), with initial condition f(0) = 1, is shown. Find limf(t.y). dt - 10 「 ンン…

A: On the slope field draw the graphical solution with initial condition f(0)=1:

Q: Is the area F = 16xi+ (2 − y2) j conserved? If the conserved field is, find the potential function.

A: Here we find the potential function by integration.

Q: Find the gradient field F = Vq for the potential function o = 3x y-y²x. %3D F = O0

A:

Q: Find a potential function f for the field F = 4y + 5z ( i + 4x j + 5x k). A general expression for…

A: Solve as general

Q: Find a potential function f for the field F. F= (y + 5z)i + (x+2z)j + (5x + 2y)k ..... A general…

A:

Q: Find a potential function f for the field F = 2y + 5z (4 i+ 8x j + 20x k). A general expression for…

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Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

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Q: Evaluate both sides for the field- rectangular parallelepiped X = 0 and 1 Y=0 and 2 and Z=0 and 3.…

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Q: The vector field A is given by A = 2r cos(r* + 3y")i + 6y cos(2² + 3y*)j. a) Show that A is…

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Q: Find the counterclockwise circulation and the outward flux of the field F = (-sin y)i + (x cos y)j…

A:

Q: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

Q: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

A:

Q: Find the potential function for the following vector field F(x, y, z) = (2x cos y – 2z3) î + (29+…

A: F = &lt;2xcosy-2z3, 29+2ye2-x2siny, y2e2-6xz&gt; To check F¯ is conservative  not we have t check…

Q: Find the gradient field F = Vo for the potential function = 2x³y-4y³x.

A: We have to find out gradient

Q: 43 Find the work done by the force field F(x, y) = (x² ln(y + 5), -) on a particle 3(y + 5) that…

A: Given force field F(x,y)=x2ln(y+5),x33(y+5) and circle x2+y2=4. The objective is to find the work…

Q: Find the potential function f for the field F = 6xi + 6yj + 4zk. %3D

A:

Q: Find a potential function f for the field F. 7y j+ y² +z? 7z 21 F = (In x+ sec(10x + 10y)) i+ sec…

A: Given that F = (ln(x) + sec^2 (10x + 10y)) * i + (sec^2 (10x + 10y) + (7y)/(y ^ 2 + z ^ 2)) * j +…

Q: Evaluate the line integral of the vector field F = xy²i+ x²yj along the curve r(t) = (t+sin,t+cos-…

A:

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

Q: Find the gradient field F = Vp for the potential function p = x®y - 4y®x. F=OD

A: Given that: φ=x6y-4y6x

Q: Find the gradient field of the function f(x.y.z) = In 4x + 4y2 +z?. ..... The gradient field is Vf=…

A: Gradient of F is given by