Consider the semicircle r(t) = a cos(t)i + a sin(t)j with 0 0. For a given vector field F, the flux across r(t) is (F· N) ds. (a) ds = a Σ dt (b) N = Σ If our vector field is F1 = 10xi – 7yj, then we have the flux being (F1. N)ds. (c) F1 (r(t)) (10a(cost))i-(7a(sint))i Σ (d) As such, (F N) ds = ((10a(cost))i-(7a(sint)))(ac E Now, if our vector field is F, = 3zi + 9(* – ylj, then we have the flux being (F2 · N)ds. (e) F; (г(+)) — Σ (f) As such, (F2· N) ds = Σ

Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+e'cos (y)) j through the right…

A: Given vector field is F=  x2y + ex siny  i + x2 + ex cosy j Curve r2=cos2θ

Q: Given the vector field E = 4zy² cos 2xa, +2zy sin 2xay + y² sin 2xa₂ for the region [x], [y], and…

A: The general form of a vector-valued function in three dimensions is given by F=F1ax+F2ay+F3az, where…

Q: Given f(x, y)= y sin(xy). find the gradient vector field くFx, Fy> =くy^2cos(xy), sin(xy)+xycos(xy)

A: We have to solve the gradient

Q: Given vector field A = 5p sinø a, - pz cosø a, + 2p a̟. At point P(1, T, -1), the vector component…

A: A= 5ρsinϕaρ-ρzcosϕaϕ+2ρaz  at point  P(1,π,-1) =0aρ-1aϕ+2az A that is parallel to  ϕ =π is…

Q: 1. Find the work done by the force field, a. F(x, y) = x²i – xyj in moving a particle along the…

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Q: 10) Find the potential function for the vector field F=-i-4j - k. z² Then compute the path integral…

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Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Compute the flux of the vector field F = xi + 5yj through the cylindrical surface as shown in the…

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Q: The following figure shows a half-cylindrical closed surface, defined by the cylinder x² + y² = 4…

A: As per the question we have to find the total flux of the vector field V through the flat top…

Q: Find the counterclockwise circulation and outward flux of the field F = xyi + 4yʻj around and over…

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Q: Find the outward flux of the field F = (x + y)î + yĵ + zk across the closed surface: S: z = 16x² -…

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Q: Problem 4: For the vector field A=xz a- yz a,- xy a, verify the divergence theorem by computing (a)…

A: Gauss divergence theorem: If F is continuously differentiable vector filed over the volume V. Then…

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: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: Find the flow of the field F = V(x²ze) once around the ellipse C in which the plane x + y + z = 1…

A: Consider the given field, F=∇x2zey around the ellipse in which the plane x+y+z=1intersects the…

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Q: Consider again the vector field F = (x +y)i + (x² + y²)j_what is the downward flux across the line…

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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: 2. Find the circulation and flux of the field F(x,y) = (-y,x) along and across the curve (cos t, 4…

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Q: Calculate the flux of the field F=xi+yj across the closed plane curve C, where C is the closed…

A: Given a field: F=x2i+y2j We need to find the flux of this field across a closed plane curve C which…

Q: Is the vector field F(x, y) = (sin(x)ely=3 + 2r cos(r>)y°, -2e)y-3 + 6 sin(x³)»*) conservative?…

A: Consider the given vector field.

Q: Consider the semicircle r(t) = a cos(t)i + a sin(t)j with 0 0. For a given vector field F, the flux…

A: Given  rt = acosti+asintj, 0≤t≤π & a>0 for a given vector f the flux across r(t) = ∫CF.Nds…

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: Find the work done by the vector field F = ( – 2, – 4y, 2æ) in moving an object along C in the…

A: Let F be a vector field defined on a smooth curve C whose parametric equation is given by rt, a≤t≤b.…

Q: Calculate the flux of the field F across the closed plane curve C where F=xi+ vj and the curve is…

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Q: The flow of a gas with a density of d = 0.3 kg/m2 987 over the closed curve r(t) = (cos t)i + (sin…

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Q: (iv) Find the work done by the force field F(x, y) = x²i+ yeªj on a particle that moves along the…

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Q: Calculate the flux of the vector field F = -2i + 5æ²j – 3k, through the square of side 2 in the…

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Q: (a) Compute the flux of the field F = r i+yj-zk across the cylindrical surface a + z = 1, 0 <y53…

A: If E be a  solid region bounded by surface of S with positive orientation then for a vector field…

Q: Q6: Find the outward flux of the vector field F=(x²y+e'sin(y))i+ (x²+ e*cos(y))j through the right…

A: Solution

Q: For Problem 2, calculate the flux of the vector field F through the given surface S. 2. F = -4i+…

A: Given F→=-4i^+xj^+2zk^ and S:x+2y+z-2=0

Q: For what values of a and d does the vector field F = ⟨ax, dy⟩ have zero flux across the unit circle…

A: A vector field, have zero flux. Across the unit-circle centered at the origin and oriented…

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

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Q: Find the work done by the force field F(x, y) = x sin(y)i + yj on a particle that moves along the…

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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: Problem 5. Use the Divergence Theorem, F. dS = div F dV, E to calculate the flux F. dS of the vector…

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Q: Find the conservative vector field for the potential function h(x,y,z) = 16xy ln(x+y) by finding its…

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Q: Calculate the flux of the field F=xi+yj across the closed plane curve C, where C is the closed…

A: theory: graph:

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 outward flux of the given field across the given cardioid. '=(3xy-1)i+(e²+5tan-'y)j 1+y…

A: Given the given expression is F=3xy-5x1+y2i+ex+5tan-1yjr=a1+cosθ, a≥0 Apply Green's theorem. Outward…

Q: d the work done by the force field F in moving a particle along the curve C. F = xyi + 3j + 9xk; C:x…

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Q: 16. Compute the flux V· nds of the vector field V(r, y) = (x² – 2ry, y² – 2xy) across the curve C :…

A: We have to compute the flux ∫ℂV·nds of the vector field Vx,y=x2-2xy,y2-2xy across the curve…

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: (а) Given the force field F = (z³ cos x + 2xy?)i+ (2x²y – 2)j + (3z² sin x – 4)k (i) Prove that F is…

A: Given force field is: F=z3cosx+2xy2i+2x2y−2j+3z2sinx−4k (i) To prove that F is conservative. (ii) To…

Q: Why does a two-dimensional vector field with zero divergence on a region have zero outward flux…

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Q: Calculate the flux of the field F across the closed plane curve C. F=xi+yj, the curve C is the…

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Q: - Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which…

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Q: Find the outward flux of the given field across the given cardioid. F=(3ry-)i+(e*+5tan-'y)j 1+y?…

A: The outward flux of given field is given by using greens theorem.

Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

Q: For the following vector field, compute (a) the circulation on and (b) the outward flux across the…

A: Counter clockwise  circulation on the region R  for the vector field F=M,Nis given by…