Consider following vector field F = (4a² + y)i + (9y³ + æ)j+ (2e*)k. Find the works done by over the following paths : • The helix 7(t) = cos(t)i + sin(t)j + k, • The piecewise smooth closed curve that is the union of paths : C1 : line segment from (1, 0,0) to (0, 0, 0) followed by C2 : the parabola z = x², y= 0 from (0,0, 0) to (1, 0, 1) followed by C3 : line segment from (1, 0, 1) to (1, 0, 0). 0

Q: Compute the flux of the vector field F = 5xi + 2yj through the cylindrical surface shown below, if a…

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Q: Consider the force field F = (2xzez — 4, cos³ z, x²eª²z – 3y cos² z sin z Find the work done by F in…

A: Given: The force field is F→=2xzex2z-4,cos3z,x2ex2z-3ycos2zsinz and the given given curve is…

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: Find the curl of the vector field F (5x sin(y), 2y cos(æ)) curl F =

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Q: 5) By using Stokes' Theorem, find the work performed by the vector field Ex² + 4xy³i+ y²xk,on a…

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Q: xi + yj x2 + y2 F(x, y) = k-

A: Topic- vector integration

Q: 2. Verify that the vector field F(1, y, 2) = sec? y+ -32 2.r tan y + e csc r, 3e -32 cot x + sec z…

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Q: Find the work done by vector field F(x, y, z) = xi + 9xyj - (x + z)k on a particle moving along a…

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Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Consider the vector field F(x, y) = yî- xî+ zR and path joining (1, 0, 1) and (-1, 0, 1) as C;:…

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Q: ) Compute the flux of the vector field F = 5xi + 2yj through the cylindrical surface shown below, if…

A:

Q: Supposing vector field G is related to the flow of fluid P is one point in the domain of G. Give the…

A: To Explain: The geometrical interpretation of G(P) where G is a vector filed and P is a point in…

Q: Consider the vector field F (x, y, z) = (y - zsinx, x, 2z + cosx). The work that performs the F…

A: Given that, F(x,y,z)=y-zsinx, x, 2z+cosxA body is displaced from point A(3π,-1,1) to point Bπ,2,0

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Q: 2. Calculate the directional derivative of the scalar field U(x, y,z) at the point M,, along the…

A: Given that : U=xz3y-x3y=xz3y-x32y12 The gradient will be calculated as :…

Q: Evaluate the line integral of the vector field (3x2 + 3y + 2xy, 3y2 + 3x + x2) along the curve y = x…

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Q: b) Find the curl of the vector field, CurlF, if F= Axzi- By 2j+ Cx tk, %3D where the coefficients A,…

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Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

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Q: Consider the vector field F = 5 sin(x² )ï + (3x² + 2e* )j. Find the line integral F· dr, where C is…

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Q: 4b. Find a "vector potential" for V, i.e. a vector field W such that curl(W): method - don't just…

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Q: Show that the vector field F=2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

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Q: What is the flux of the vector field (x2,y2,z1 ) through a cuboid with x ranging from 0 to 8.3, y…

A: use divergence theorem

Q: Is the vector field F(x, y) = (6ry cos(3x²y + e*) +e* cos(3x?y + e*), 3x² cos(3x²y + e*))…

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

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Q: Graph the vector field F(x, y)=(-3y, 2x); also find the divergence and the rotational.

A: The graph of the  vector field F=-3y, 2x is as follows

Q: compute dr for the vector field where Cis the line segment from Co.o) to Cli3)

A: Given vector field is, F=1+x2,xy2  The objective is to compute ∫CF·dr, where C is the line segment…

Q: Find the curl of the vector field F = (4x sin(y), 7y cos(x)). curl F - Dretieur

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Q: Compute the flux of the vector field F = 5xi + yj through the cylindrical surface shown below, if a…

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Q: b) Find the curl of the vector field, CurlF, if F = Ax²z²i– By:'j+Cz*k, where the coefficients A,…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Find the curl of the vector field F. F(x, Y, 2) = arcsin yi+ V1 - xzj + yêu y²k

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Q: Consider the following vector field: F = (y* + 1, 3ry + 1). (8.1) Is ſ. F · dr path-independent?…

A: The given vector field is F=y3+1, 3xy2+1. (8.1) For a vector field F=P,Q, the line integral ∫CF·dr…

Q: A conservative vector field is given as F = 6xe'i + (3x²e" –10yz')j-15y²z°k. Find the potential…

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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: 3. Find the work of the vector field F(2, y, ) = yi along the curve which is obtained as the…

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Q: Consider the vector field F = ⟨ 1 , y ⟩ .Compute the path integral of this field along the path:…

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Q: A=(2x-3y)ax+(2xy-y2)ay Prove the Stokes theorem for the triangular linear path in the shape that is…

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Q: he temperature in a temperature-controlled chamber is given by a 3D scalar field Tlx, y, 2) = Ax®yz…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Calculate the divergence and the curi of the following vector field at the point (1. 2. 1) Fix, y,…

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Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…

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Q: 5) Integrate the vector field h a,y)= y2i+ (2xy-e*)J C parumetrizal by t€ £0, 5 3. over the circular…

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Q: Let C be the curve of intersection of z = 1 – 2x2 and the cylinder x2 + y2 = 1 oriented…

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Q: Compute the curl of the vector field F = 2y3 i+ e j+ cos(x) k curl

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Q: Determine the curl of the vector field A = pz sin o ap + 3pz cos o ao at (5, ,1) %3D O-1 O -6pz cos…

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Q: Find the counterclockwise circulation and outward flux of the field F = 7xyi + -5y²j C enclosed by…

A: Introduction: Let C be a piecewise smooth, simple closed curve enclosing a region R in the plane.…

Q: Consider the vector Field 7 and the curve C below. F(x,y) = (3 +4 ху?) 1 + 4x²yј, C is the arc of…

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Q: Sketch the vector field F(x, y) = 3i + 5j and calculate the flux then the circulation of F| along…

A: (1) The equation of line segment is: x+3-2+3=y-1-5-1=tx+3=t⇒x=t-3y-1=-6t⇒y=-6t+1 The curve…

Q: 4. Given the vector field Č = p² sin o a, + (z² cos – 1)as + (p² – z)a, Determine the x-component of…

A: Given the vector field

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.