Question

vector calculus.

Do both please

Q1: What, in your own words, is the difference between a partial derivative and a directional derivative? How are they similar? Give a particular example to illustrate your explanations (choose some function z=f(x,y)) Q2: Given a surface z= f(x,y), a point (x,yo) in the domain of f, and a unit vector i pointing some direction in the xy-plane, what does it mean if D,f(x,Y)=0? Be as specific as possible.

Answer 1

1. Directional derivative is the instantaneous rate of change (which is a scalar) of f(x,y, z) in the direction of the unit vector u. The directional derivative is a number; it is the rate of change when your point in R³ moves in that direction. (You can imagine "reducing" your function to a function of a single variable, say tt, by "slicing" the curve in that direction; the directional derivative is just then the 1-D derivative of that "sliced" function.)

Partial Derivative is the rate of change of f(x,y, z), along x, y and z directions, which can be thought of the slope of the function at a point (x₀, y₀, z₀). The gradient is a vector; it points in the direction of steepest ascent.

In simple words, directional derivative can be visualized as slope of the function at the given point along a particular direction. For example partial derivative w.r.t x of a function can also be written as directional derivative of that function along x direction. So, the gradient is a vector with the slope of the function along each of the coordinate axes whereas the directional derivative is the slope in an arbitrary specified direction.

2. If D_uf(x₀, y₀)=0, then f is constant in the direction u i.e., if we consider the values of f along the line in the direction u, we find that there is no linear-order term:

f(x₀+t u, y₀+s u) = f(x₀, y₀) + t D_uf(x₀, y₀ + s u) + s D_uf(x₀ + t u, y₀) +O(t, s) = f(x₀, y₀) + O(t, s).

(Here O(t, s) is some function such that O (t, s) → 0 as t→0, s→0.)

Instead of considering a line / plane, we can consider the level set . So long as (x₀, y₀) isn't a critical point of f, this level set will (at least in some neighborhood of (x₀, y₀) be a curve. Thus, f is constant along a curve in the direction u.