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: 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: Determine the vector field of FF. F(x,y)=8i−4j

A: Solution

Q: If the gradient of f is Vƒ = z j - yi – 2xzk and the poi f(x, y, z) = 0, find an equation for the…

A:

Q: Given the vector valued function: T(t) = Then the point 9 units along this curve (in the direction…

A: Tangent vector two a curve always point in the direction of t increasing.

Q: The directional derivative of f(x, y) = cos(x²y³ ) in the direction of the vecto v = 3i + 4j is Du…

A: The given function is: f(x,y)=cos(x2y3) So, ∇f(x,y)=∂f∂x,∂f∂y…

Q: 2- Calculate the divergence of the following vector functions: (a) V = (r cose) r + (r sine) 0 + (r…

A:

Q: Find the derivative of the vector function r(t) = ta x (b+ tc), where a = (4, -1,5), b = (-3,-3,-1),…

A: Given, the vector function      r(t)=ta×(b+tc),where       a=4,-1, 5, b=-3,-3, -1, and  c=1, 2, 2,…

Q: Let u(t) = 2t°i + (? -7)j-8k and v(t) = e'i+3 ej- ek. Compute the derivative of the following…

A: