JNTU B.Tech II MATHEMATICS-II Previous Question Paper (July/August - 2021 - R13)

A scalar field F = P(x, y)i + Q(x, y)j is conservative if ∂Q/∂x = ∂P/∂y. In this case, P = x² + xy² and Q = y² + x²y.

First, we find the partial derivatives:

Since ∂Q/∂x = ∂P/∂y, the field F is conservative.

To find the scalar potential function ϕ, we know that ∇ϕ = F, which means:

Integrating the first equation with respect to x:

Differentiating this with respect to y:

Equating this to Q:

This gives g'(y) = y².

Integrating g'(y) with respect to y:

Substituting g(y) back into the expression for ϕ(x, y):

Thus, the scalar potential function is ϕ(x, y) = x³/3 + x²y²/2 + y³/3 + C.

The directional derivative of a scalar function f(x, y, z) at a point (x₀, y₀, z₀) in the direction of a unit vector u is given by D_u f = ∇f ∙ u.

The given scalar function is f(x, y, z) = x²yz. The point is (1, 2, -1).

First, we find the gradient of f:

Calculate the partial derivatives:

So, ∇f = (2xyz)i + (x²z)j + (x²y)k.

Evaluate the gradient at the point (1, 2, -1):

The direction is given by the vector v = i + 2j + 3k. We need to find the unit vector in this direction:

Now, calculate the directional derivative:

The directional derivative of x²yz at (1, 2, -1) in the direction of i+2j+3k is 0.