Along the curve C traced out by the end point of the position vector tk (t) = (- sin ti + cos tj+t) = [0,π/2] calculate the line integral √ F(r) - di dr of the vector field F(x, y, z) = −y+xj+yzk.

Q: Find the directional derivative of the scalar field = x¹y + 4xz at the point (2, -2, 4) along the…

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Q: Find the directional derivative of the scalar field ¢ = xªy + 3xz at the point (1, −1, 4) along the…

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Q: For the vector field G = (yey + 3 cos (3x + y))i + (xeªy + cos(3x + y)), find the line integral of G…

A: We find the required integrtion using fundamental theorem of line Integral.

Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 3) to (9, 3), and the line segment…

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Q: Define two vector functions 7(t) = 2 sin(t)i + 8 cos(t)j + t°k ü(t) = 8 sin(t)? + 2 cos(t)j + (t? –…

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Q: 2. Find the curl of the vector field. F = z°e* sin(2 y)i – e' cos yj+ xyzk to

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Q: Suppose F(x, y) = x² + y² and C is the line segment from point A = (1, -1) to B = (3,-5). (a) Find a…

A: Given          Let       (a)   The equation of the line segment from A(1, -1) to B(3, -5) is,,…

Q: Consider the vector field v(x)=(y/(x^2+y^2))i − (x/(x^2+y^2)) j + 4k. Calculate ∫Cv dr, where C is…

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Q: For the vector field G = (yety + 4 cos(4x + y))i + (xey + cos(4x + y)) 5 I find the line integral of…

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Q: The gradient of f(x, y) = x sin (y²) is the vector field F (x, y) = at (2,√π/6) and ending at…

A: Here we are given that the gradient of f(x,y)=xsin(y2) is the vector field F(x,y)=sin(y2)2xycos(y2)

Q: A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at…

A: A sphere of radius 1 is x2 + y2 + z2 = 1. Parametrize the sphere as follows:Now,calculate the…

Q: Find the curl of the vector field F = (3y cos(x), 5x sin(y))

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Q: Show that the vector-valued function shown below describes the function of a particle moving in a…

A: Part - 1 : Given vector-valued function is                                      rt = 5i +5j +3k +…