Engineering Mat...

If \(\vec R\left( t \right)\) is a vector having constant magnitude, then the correct statement is

Which of the following is/are true?

1. \(\nabla \times \left( {\nabla \times {\rm{\vec V}}} \right) = 0\)

2. \(\nabla \times \left( {\nabla \times {\rm{\vec V}}} \right) = \nabla \left( {\nabla .{\rm{\vec V}}} \right) - {\nabla ^2}{\rm{\vec V}}\)

3. \(\nabla .\left( {\nabla \times {\rm{\vec V}}} \right) = 0\)

A triangle defined by A(2, -5, 1), B(0, 2, 4) and C(0, 3, 1). What is area of the triangle?

Consider an incompressible flow velocity given as

\(\vec V = \left( {2x + 3y + 4z} \right)\hat i + \left( {5x + cy + 6z} \right)\hat j + \left( {8x + 9y} \right)\hat k\) 

The linear velocity profile at any section is given as-

\(\vec V = \left( {2{x^2}y} \right)\hat i + \left( {xy{z^2}} \right)\hat j + \left( {xy} \right)\hat k\)

If f(x, y, z) = x²y + y²z + z²x for all (x, y, z) ϵ R³ and \(\nabla = \frac{\partial }{{\partial x}}i + \frac{\partial }{{\partial y}}j + \frac{\partial }{{\partial z}}k\), then the value of ∇.(∇ × ∇f) + ∇.(∇f) at (1, 1, 1) is

If the function F is solenoid and \(\rm \left| {\vec F} \right| = {r^{n - 1}}\), then value of \(\rm n\) will be –

For a position vector \(\vec r = x\hat i + y\hat j + zk\) the norm of the vector can be defined as \(\left| {\vec r} \right| = \sqrt {{x^2} + {y^2} + {z^2}}\). Given a function \(\phi = \ln \left| {\vec r} \right|\), its gradient ∇ϕ is