= (-2y)i + (x)j – (3)k and S is the Vx2 + y2 , 0

Q: 8. Find the flow of the velocity field F(x, y) = ( + y)î – (x² + y² )j along the circle a? + y² = 1…

A: According to the given information, it is required to find the flow of velocity field.

Q: Verify Stokes' theorem for F = (x+y-4) i+3xy j+ )k, over the surface of the hemisphere x2 + y2 + z2…

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Q: Find an equation of the tangent plane to the given surface at the specified point z = sin(xy), (x,v)…

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Q: Calculate the integral of f(z,y, 2) = 6z2 + 6y² + 2³ over the curve c(t) = (cost, sin t,t) for 0<t<…

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Q: Let F = (-z, x, y²). Find curi Fds, where S is the paraboloid z = x2 + y?, 0 < z< 4. (You can do the…

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Q: I = y dx + (z — х) dy — ydz, Jc

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Q: Find the tangent plane of the surface z = f(x, y) at (0,0, 0) in which 3y f(2, y) = || cos /3 t dt.

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Q: Find the coordinates of the centroid of the curve x = e^t cos t, y = e^t sin t, 0&lt;=t&lt;= pai.

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Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

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Q: Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate…

A: Concept: Similar to circle sphere is a two dimensional space where the set of points that are at the…

Q: Use V(P) = k / Kx, y, z2) ds Dp(x, y, z) where k = 8.99 x 10° N - m²/C°, to compute the electric…

A: Consider the provided question, VP=k∫C∂x,y,zDpx,y,zdswhere k = 8.99 x 109 N·m2c2. To compute the…

Q: Use a scalar line integral to find the length of the curve r(t) = (20 sin , 20 cos ,5), for 0 < t <…

A: Given

Q: Use a parameterization to find the flux | F.n do of F = 5xy i- 4z k outward (normal away from the…

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Q: Let v = zk be the velocity field (in meters per second) of a fluid in R3. Calculate the flow rate…

A: volume ( V)= 2/3pi r3 radius of sphere in vector form can be written as = x i^ + y j^ + z k^ so…

Q: CurlF dS x+y' +z =9, z20 se Stokes' Theorem to compute s for the hemisphere intersected with the xy-…

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Q: Find equations for (a) the tangent plane and (b) normal line at the point P on the given surface 1)…

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Q: 10. Use Stokes' Theorem to evaluate ||(V×F)· dS where F= (z,2x, y) and Sis the hemisphere of x' + y?…

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Q: Evaluate the integral |xy dx – z dy + 8z dz where C is the curve with parametric equations x = 2t, y…

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Q: Find the coordinates of the centroid of the curve x = cos t + t sin t, y = sin t - t cos t, 0 … t ……

A: coordinates of the centroid of the curve  x=cost+tsinty=sint-tcost…

Q: Given the scalar function f(x,y)=2xy,calculate the line integral along the ellipse curve from point…

A: Given data:- The given curve is, x29+y24=1 The given scaler function is, f(x,y)=2xy t E 0,π/2

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: Integrate f(r, y, z) = for 0 <t< V3. + y² over the curve r(t) = [cos(t) +t sin(t), sin(t) – t…

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

A: Use tangential form of Green's Theorem to find the counterclockwise circulation. Counterclockwise…

Q: Suppose F = ∇ * A, where A = (y + sqrt(z))i + e^(xyz) j + cos(xz) k. Determine the flux of F outward…

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Q: Use polar coordinates to evaluate the double integral V1 1-z2 dy dæ. (1+z² + y?)²

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Q: S,F•dr, and C is the triangle, with Use Stokes Theorem to evaluate where F(x,y; =) = (x+ y² )i+ (vy+…

A: Given: Fx,y,z=(x+y2)i+y+z2j+z+x2k and C is the triangle with vertices 1,0,0 ,0,1,0 and 0,0,1 We know…

Q: Verify Stokes' Theorem for F(x, y, z) = xi + yj + y²zk where S is the first octant portion of the…

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Q: 1. Compute ff,(V × F) - dS, where S is given by r² + y? + (z – 1)? = 2 and z > 0, and F(x, y, z) =…

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Q: - Evaluate the integral 4-z²-y² 1 dz dy dx Jo 4-z²-y² x² + y² + 2² using spherical coordinates.

A: To evaluate ∫-22∫04-x2∫-4-x2-y24-x2-y2 1x2+y2+z2 dzdydx

Q: 1. Compute ff(V × F) - dS, where S is given by r² + y²+ (z – 1)² = 2 and z > 0, and F(r, Y, z) =…

A: Given: S is given by x2+y2+z-12=2  and z≥0 Fx,y,z=-esinzy,x+sinz,cosx

Q: The surface area of the solid of revolution generated by rotating x = (y² + 2)²,1 < y s 2 about the…

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Q: Use cylindrical coordinates to evaluate the triple integral /// (æ – y) dV, where E is the solid…

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Q: For a certain fluid flow field the velocity is given by V = [y² – x(1+ x)]i+ [y(2x + 1)]j+k. Show…

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Q: Use Stokes's Theorem to evaluate F(x, y, z) = z²i+yj + zk S: z = √4x² - y2 F. dr. In this case, C is…

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Q: Evaluate /|/ (x² + y² + z²)² dV where E is the solid ball centered at the origin with radius 2.

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Q: 4. Integrate f(x, y, z) = Vx² + y² over the involute curve (cos t + t sin t)i + (sin t – t cos t)j,…

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Q: Use Stokes' theorem to evaluate V ×F· î dS where = (9y²z, 4xz, 2x²y²) and S is the paraboloid z = x²…

A: Step:-1 Stoke's Theorem:- ∬s∇×F→.n dS=∫F.dr Now, given that F→=9y2z, 4xz, 2x2y2 and paraboloid…

Q: Calculate the integral ∫C(x2+y2)zds, where the curve C is parameterized by r(t)=(sin3t,cos3t,4t),…

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Q: dw Let w =xycosz, where x = t°, y =ť, and z = arccost. Find dt

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Q: Calculate the work done by F(x,y) = ⟨2x + ln( 2y2 −1), arctan(y4 +2y +1) + 4xy / (2y2 − 1⟩ along the…

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Q: valuate the line integral J xev% ds where C is the line segment from > (1, 1, 1). This yields:

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Q: Show that the curve r(t) = (ln t)i + (t ln t)j + t k is tangent to the surface xz2 - yz + cos xy = 1…

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Q: 2. Use Stokes' Theorem to evaluate curlF.dS whereF = 2y i + 3z j+(z – z) k and S is the portion of y…

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Q: Let F = (5x4 + 16z, 2x³ + 3y? + 16z, x + y+ ev*+1) and C be the curve defined by 7 (t) = (2 cos(t),…

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Q: Let F = ( ) 5x* + 16z, 2x³ + 3y² + 16z, x + y + eVz³+1 and C be the curve defined by 7(t) = (2…

A: Given that,  F=5x4+16z, 2x3+3y2+16z, x+y+ez3+1 And rt=2cost, 2sint, 16cos2t sin2t, o≤t≤π To find…

Q: If F(XN.2) = tan z į+ (xtz)j + x sin y k and o is the portion of the cone z=2-Vx²+y? above the…

A: The Stokes's Theorem states that, the surface integral of the curl of a function over a surface…

Q: Calculate Sf(F(t)) | (t)) dt f √ √ √x + 1 (x+y+z) ds where C is the helix r(t) = (cos(t), sin(t),…

A: What is Line Integral: A line integral in mathematics is an integral in which the function to be…

Q: Define stokes theorem and evaluate the line integral SF.r'(s) ds where F = [4z, 2x, 2x], C: the…

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Q: Use Stokes' Theorem to evaluate |/. CurlF dS, where F(r, y, 2) = xyzi + xj + e" cos zk and S is the…

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Q: 3. Integrate f(x, y, z) = Vx² + z² over the circle (a cos t)j + (a sin t)k, r(t) r(t) 0 <t< 2m.

A: