3. Consider a square C in the xz plane of R³ with vertices (0,0,0), (1,0,0), (1,0, 1), (0,0, 1), oriented counterclockwise as seen from the positive y-axis, and a vector field F(x, y, z) given by F(x,y, 2) = (x +z + a) i+ (3a) j + Gx+ 2) k whose components depend on some real value a € R. (a) For which value of a is F a conservative vector field? (b) For that value of a, compute the line integral fc E · dr without doing any explicit integrations. Explain your reasoning. (c) Calculate the line integral ſc E • dr for a = 0.

Q: c) Show that the vector field W(x, y, z) = (2yz+ xz – x², –2xz – yz + y?, y² – x² + z²) - - is a…

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Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: 1) Find the work done by the vector field F(x, y, z) = (x², 4xy³, xy²) on a particle that %3D…

A: Given force is F→x,y,z=x2,4xy3,xy2 Let r→=rt→ be the boundary of the rectangle with vertices  0,0,0,…

Q: Compute the vector assigned to the points P = (1, 1) and Q = (-3, -3) by the vector field F = (-y,…

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Q: Set V = {(x, y) | x ER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x')…

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Q: Consider the vector field v(x)

A: Given, the vector field        v(x)=yx2+y2i^-xx2+y2j^+6k^                               ...(1)

Q: Let c1 (t) = ei + 7 sin(t)j + t*k and c2 (t) = e-Sti + 5 cos(t)j – 7t*k. (Enter your solution as a…

A: Given: Let c1(t)=e5ti+7sintj+t4k and c2(t)=e-5ti+5cos(t)j-7t4k. To Find:      ddt[c1(t)+c2(t)]

Q: Let C₁ be the directed line segment that starts at (2,0) and ends at (-3,-5) and C₂ be the parabola…

A:

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

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Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: 2. For the vector field F(x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div…

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Q: (b) In the (x, y)-plane, sketch the vector fields v = (y, 2y, 0) and w = (x−y, x+y, 0), clearly…

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Q: Date: الموافق Deternine where the function isut Contineys.

A: A function is said to be continuous function, if the function is defined at all points of x and the…

Q: d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V…

A: V(x,y,z) = <2x+3y,2y+3x,-4z>F={ (x,y,z) : z=4-x2-y2 , z≥0}Consider, g(x,y,z) = z+x2+y2-4…

Q: 13. For the vector fields: A = x + 2y ŷ + 3z 2; B = 3y - 2x ŷ. Check both product rule given in…

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Q: magnitude 1.

A: To find : (x0,y0) Given: (x1,y1) = (2,1) Magnitude = 1

Q: Q5) If the vector field T = (axy + Bz³)a, + (3x² – yz)a, + (3xz² – y)a, is irrotational, determine…

A:

Q: Consider the vector field F(x, y, z)  =  (Sx3y3z7)i + (3x4y2z7)j + (Tx4y3z6)k Find the values of S…

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Q: x-z+4 8x-y+3z² 3. Given vectors field G = 3* 4+z2 :ax + y3+z - ay and a position vector, r, = 2ax –…

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Q: What is the curl of the vector field F(z, y, z) = ryi+ zyzj + yzk O a. (-ry + z)i + (yz – 2)k O b.…

A: We have to find

Q: Consider the square C oriented counter clockwise, with vertices (0, 0), (2, 0), (0, 2), (2, 2) and 7…

A:

Q: 4. Match the vector fields given below with their plots. (a) F(x,y, z) = (b) F(x, y, z) = (c) F(x,…

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Q: Let F₁ (x, y, z) = (y, x-z²+2, -2yz) and F₂(x, y, z) = (x − z², -y +2, 2yz). (a) Find the curl of…

A: Given vectors  and  we need to find the curl of each vectors and then determine which vector is…

Q: Given two vector fields E = 5xya, + 6(x + 3)a, + 8z'a, D = yza, + xza, + xya, (a) Evaluate C= D+E at…

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Q: Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.

A: Given the vector field function u(r)=(y2,x+y,0). We need to plot the vector field in the…

Q: Let a vector field F(x,y)=sinx 1 + xy j be given. What is the integral representing the work done by…

A: Acoording to expert guidelines, I am allowed to answer first question only. Kindly repost the rest…

Q: Sketch the given vector fields by drawing some representative vectors. (a) F(x, y) = (x − y) î+xî…

A: Given:  a F→x, y=x-yi^ +xj^b F→x, y=y, -xx2+y2c F→x, y=2yj To sketch the given vector field.