Consider a function
$$f(x,y,z,w) = x^2 + 2y +3 \sin z +4w^3 $$
Then the gradient vector is given by
$$\nabla f = [f_x, f_y, f_z, f_w] = [2x, 2, 3 \cos z, 12w^2]$$
But, is it the same as a gradient vector field, or gradient vector field is a set of other vectors defined by $f$
i.e., $\vec{f} = 2x \hat{i} + 2 \hat{j} + 3 \cos z \hat{k} + 12w^2 \hat{l}$ ?