Q1) Determine the gradient of the following scalar field V = e(2x+3y)cos5z Q2) Given f = x²y + 3z + 4, find (i) Vf. (ii) V²f, and (iii) a unit normal to f = 0 at (1, -2, 4). Q3) Find the divergence and curl of the following vectors: A = ey a, + sinxy a, + cos? xz a, Q4) Given the vector field R = (2xły + yz)a, + (xy² – xz')a, + (cxyz – 2x²y²)a, Determine the value of c for R to be solenoidal. (Note: solenoidal means the divergence equal to zero) Q5) If the vector field T= (axy + Bz³)a, + (3x² – yz)a, + (3xz² - y)a, is irrotational, determine a, B, and y. Find V-T at (2, -1, 0). (Note: irrotational means the curl equal to zero)

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

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Q: Define two vector functions F(t) = 8 sin(t)i + 6 cos(t)j + (t – 2)k ü(t) = 6 sin(t)i + 8cos(t)j + (t…

A: We have to find the dot product of r(t) and u(t)

Q: x-y, z + y + 3,z²) and the net is decribed by the equation y = 1- x² - z on

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Q: Given r(t) be a vector function that satisfies: r'(1)= T'(1)= 1) Determine the unit tangent…

A: Given, r'(1)=<-3,0,4> T'(1)=<2,2,1>

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: Find the difference c-d for the following cubic spline S) with natural boundary conditions: S(2) = {…

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Q: The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end…

A: Given, vector field F→x,y=2xy,x2 and three curves that start at 1,2 and end at 3,2.

Q: 2.31 Let A and B be vector functions of position vector x with continuous first and second…

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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: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…

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Q: 1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction…

A: Step 1:

Q: F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?

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Q: Q1: a) Sketch the vector field and find the fixed points of the following function i = x - V1 -…

A: Given equation is x.=x-1-sin2x and we have to find the fixed points of this equation. We will use…

Q: 7. For the vector-valued function √t-1 ln(2t - 1) t² - 1 r(t) = (VI- t-1 2 COS πt rt), (t (t = 1):…

A: Let us solve above problem

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: Let 7 (t) = (4t³ – 4, – 3e", – 3 sin( – 4t)) - Find the unit tangent vector T(t) at the point t = 0…

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Q: (a) Find the direction for which the directional derivative of the function f(x, y) = 9xy + 2y is a…

A: Formula for directional derivative Du→fP=∇f(P).u^=∇f(P)u^cosθ Du→fP to be maximum angle θmust be 0.…

Q: The gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² »i O II. (4xy)i…

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

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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: 1) Sketch the vector field F. (i) F(r, y) = (ii) F(r,y, 2) = i

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