The gradient of ϕ=3x2y−y3z2𝜙=3𝑥2𝑦−𝑦3𝑧2 is

The gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

The scalar field given is ϕ=3x²y−y³z².

The gradient of a scalar field ϕ in three dimensions is given by the vector (∂ϕ/∂x, ∂ϕ/∂y, ∂ϕ/∂z).

So, we need to compute the partial derivatives of ϕ with respect to x, y, and z.

∂ϕ/∂x = ∂/∂x (3x²y−y³z²) = 6xy

∂ϕ/∂y = ∂/∂y (3x²y−y³z²) = 3x² - 3y²z²

∂ϕ/∂z = ∂/∂z (3x²y−y³z²) = -2y³z

So, the gradient of ϕ is (6xy, 3x² - 3y²z², -2y³z).