Let  ϕ(x,y,z)=3x2y−y3z2.𝜙(𝑥,𝑦,𝑧)=3𝑥2𝑦−𝑦3𝑧2. Then the gradient of ϕ𝜙 at (1,1,0) isa.6i^−3j^+0k^6𝑖^−3𝑗^+0𝑘^b.None of thesec.6i^+3j^+0k^6𝑖^+3𝑗^+0𝑘^d.6i^+6j^+0k^6𝑖^+6𝑗^+0𝑘^Clear my choice

The gradient of a scalar field ϕ(x,y,z) is a vector field that points in the direction of the greatest rate of increase of ϕ, and whose magnitude is the greatest rate of change.

The gradient of ϕ is given by the vector (∂ϕ/∂x, ∂ϕ/∂y, ∂ϕ/∂z).

Let's compute these partial derivatives:

∂ϕ/∂x = ∂/∂x [3x^2y - y^3z^2] = 6xy ∂ϕ/∂y = ∂/∂y [3x^2y - y^3z^2] = 3x^2 - 3y^2z^2 ∂ϕ/∂z = ∂/∂z [3x^2y - y^3z^2] = -2y^3z

Now, let's evaluate these at the point (1,1,0):

∂ϕ/∂x (1,1,0) = 611 = 6 ∂ϕ/∂y (1,1,0) = 31^2 - 31^20^2 = 3 ∂ϕ/∂z (1,1,0) = -21^3*0 = 0

So, the gradient of ϕ at (1,1,0) is (6,3,0), which corresponds to 6i^ + 3j^ + 0k^. Therefore, the correct answer is c. 6i^ + 3j^ + 0k^.