Engineering Mat...

Divergence value of a function \({x^3}y\vec i - \left( {{z^2} - 2y} \right)\vec j + 5{y^2}z\vec k\) at x = 2, y = 3 and z = 4 is

If \(\bar r = x\hat i + y\hat j + z\hat k\) and r = |r̅|, then ∇² [log r] = ?

Consider an incompressible flow velocity given as

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

Let \(\vec a = \lambda \hat i - 9\hat j - \hat k,\;\vec b = 3\hat i + 3\hat j + \hat k\;and\;\vec c = 4\hat i + 2\hat j + \hat k\). The value of λ for which the vector \(\vec a\) is perpendicular to \(\vec b \times {\rm{\;}}\vec c\) is ________.

The value of \(\mathop \smallint \limits_C \left[ {\left( {2{x^2} - y} \right)dx + 3x{y^2}dy} \right]\) and C is the curve x² = 4y and y² = 4x is _____.

The circulation of A̅ = yî + zĵ + xk̂ around the circle x² + y² = 1, z = 0 is

If f = x²yz and g = xy – 3z², then the value ∇ ⋅ (∇f × ∇g) of (1, -1, 2) is ______ 

The vector \(\vec V = \left( {x + y + az} \right)i + \left( {bx + 2y - z} \right)j + + \left( { - x + cy + 2z} \right)k\) is irrotational. Where a, b and c are constants. Find the divergence of the vector \(\vec V\).

The directional derivative of ϕ = x²yz + 4 xz² at (1, -2, -1) in the direction 2î - ĵ - 2k [upto 2 decimals]

If F = 3y î – xz ​ĵ + yz² k̂ and S is the surface of the paraboloid 2z = x² + y² bounded by z = 8, evaluate \(\mathop \int\!\!\!\int \limits_s^\; \left( {\nabla \times F} \right).ds\)