Derivatives and Differentials

References are to Salas/Hille/Etgen’s Calculus, 8th Edition (Wiley, 1999)

We wish to differentiate a scalar-valued function f(r) of a vector variable r at the point P_{0} = r_{0}. For simplicity, we assume r is 2-dimensional and write r_{0} = (x_{0}, y_{0}); however, everything extends without difficulty (except notationally) to n dimensions.

Differentials We want our basic definitions to be coordinate-free so as to ensure some geometric content. We reduce to a one-dimensional problem by restricting attention to the values of f along the general line through r_{0} with direction vector h = h_{1}i + h_{2} j, which may parametrized by r(t) = r_{0} + th, by setting

g(t) = f(r(t)) = f(r_{0} + th);  (1)

this function is defined for all small t (if r_{0} is an interior point of the domain of f).

Definition 2 Given the point r_{0} and the vector h, we define the differential df of f by

df(r_{0}, h) = g′(0) = lim_{t→0} \frac{g(t) − g(0)}{t} = lim_{t→0} \frac{f(r_{0} + th) − f(r_{0})}{t},  (3)

provided that this limit exists.

Note that here h can be any vector; it does not have to be small and it does not have to be a unit vector.

Directional derivatives If h is a unit vector, df(r_{0}, h) is also called (see Defn. 15.2.2) the directional derivative of f at the point r_{0} in the direction h, and is sometimes written f′_{h}(r_{0}) or D_{h}f(r_{0}). (We shall not use either notation.) But if h is not a unit vector (and not zero), we can replace h by the unit vector (1/∥h∥) h and call df(r_{0}, (1/∥h∥) h) the directional derivative of f in the direction of h (see the Remark on p. 882). This usage in the book is confusing and is not recommended.

Partial derivatives For purposes of computation, we are particularly interested in the two coordinate directions, i and j. It is convenient to write f(r) = f(x, y) here.

Definition 4 (cf. Defn. 14.4.1) The partial derivative f_{x}(r_{0}) or \frac{∂f}{∂x}(r_{0}) of f with

respect to x at the point r_{0} = (x_{0}, y_{0}) is the directional derivative

f_{x}(r_{0}) = f_{x}(x_{0}, y_{0}) = \frac{∂f}{∂x}(r_{0}) = df(r_{0}, i) = lim_{t→0} \frac{f(x_{0} + t, y_{0}) − f(x_{0}, y_{0})}{t}. (5)

Similarly, the partial derivative of f with respect to y at r_{0} is

f_{y}(r_{0}) = f_{y}(x_{0}, y_{0}) = \frac{∂f}{∂y} (r_{0}) = df(r_{0}, j) = lim_{t→0} \frac{f(x_{0}, y_{0} + t) − f(x_{0}, y_{0})}{t}. (6)

If u = f(r), one often writes u_{x} for f_{x} and \frac{∂u}{∂x} for \frac{∂f}{∂x}.

Linearity It is easy to see that df(r_{0}, kh) = k df(r_{0}, h) for any constant k. (Changing h to kh replaces g(t) in equation (1) by g(kt).) However, the values of f along different lines through r_{0} are quite independent, apart from the obvious requirement g(0) = f(r_{0}). This makes it quite easy to manufacture all kinds of bizarre behavior of df; see the book for some examples. (The existence of df does not guarantee that f is continuous at r_{0}, see Example 2 on p. 860, or even that f is bounded near r_{0}.)

Nevertheless, such pathology rarely occurs in practice. Instead of moving from P_{0} to P (= r_{0} + h = (x_{0} + h_{1}, y_{0} + h_{2})) directly, we can go by way of Q (= (x_{0}, y_{0} + h_{2})); from P_{0} to Q, we change only the y-coordinate, and from Q to P we change only the x-coordinate. One often writes Δx for h_{1} and Δy for h_{2} here. If the partial derivatives of f exist at all points near r_{0}, we can estimate f(P) − f(P_{0}) = (f(P) − f(Q)) + (f(Q) − f(P_{0})) by using the one-dimensional Mean-Value Theorem, in the form

g(t + h) − g(t) = g′(t + θh) h  (7)

for some θ satisfying 0 < θ < 1, assuming that g′ exists.

Theorem 8 (cf. Thm. 15.1.3) Assume that f_{x} and f_{y} are continuous at r_{0} (and therefore exist at all points near r_{0}). Then f is differentiable at r_{0},

f(r_{0} + h) = f(r_{0}) + f_{x}(r_{0}) h_{1} + f_{y}(r_{0}) h_{2} + o(h) as h → 0.  (9)

Proof See the book, p. 878. For the notation o(h), see p. 872.

Under these mild hypotheses, we easily compute the differential df.

Theorem 10 If f_{x} and f_{y} are continuous at r_{0}, the differential df is given by

df(r_{0}, h) = ∇f(r_{0}) • h = f_{x}(r_{0}) h_{1} + f_{y}(r_{0}) h_{2} = \frac{∂f}{∂x}(r_{0}) h_{1} + \frac{∂f}{∂y} (r_{0}) h_{2}.  (11)

Proof If we use equation (9) (with h replaced by th), equation (1) becomes

g(t) = f(r_{0} + th) = f(r_{0}) + f_{x}(r_{0}) th_{1} + f_{y}(r_{0}) th_{2} + o(t) as t → 0. Now we can read off g′(0) as required. The term o(t) contributes nothing.

Approximation If we plug equation (11) back into equation (9), we obtain

f(r_{0} + h) = f(r_{0}) + df(r_{0}, h) + o(h) as h → 0,  (12)

which shows how the differential df gives a useful approximation to f(r_{0} + h).

The coordinate differentials The coordinate x can itself be considered a function of r, and its differential is readily computed from equation (11), or directly from equation (3), as dx(r_{0}, h) = 1h_{1} + 0h_{2} = h_{1}. Similarly, dy(r_{0}, h) = h_{2}. This allows us to rewrite equation (11) as the equation of differentials

df(r_{0}, h) = f_{x}(r_{0}) dx(r_{0}, h) + f_{y}(r_{0}) dy(r_{0}, h).  (13)

If we suppress all the evaluations and switch to the other notation, this takes the traditional and memorable form (cf. (15.7.4))

df = \frac{∂f}{∂x}dx + \frac{∂f}{∂y} dy.  (14)

So df/dx is not the same as ∂f/∂x, unless dy happens to be zero.