Theory

Differential calculus in several variables

Partial derivatives and gradient, differentiability and tangent plane, directional derivatives and the gradient formula, chain rule, second derivatives and second-order Taylor formula.

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Partial derivatives and gradient
Differentiability

Partial derivatives and gradient

1. Partial derivative

$f : A \subseteq R^{2} \to R$ , $(x_{0}, y_{0})$ interior to $A$ . The partial derivative of $f$ with respect to $x$ at $(x_{0}, y_{0})$ is

\frac{\partial f}{\partial x} (x_{0}, y_{0}) = t \to 0 lim \frac{f ( x _{0} + t , y _{0} ) - f ( x _{0} , y _{0} )}{t},

if the limit exists and is finite; the partial derivative with respect to $y$ is defined in the same way, with the increment on the second variable. One also writes $f_{x} (x_{0}, y_{0})$ and $f_{y} (x_{0}, y_{0})$ . If both exist, $f$ is called partially differentiable at $(x_{0}, y_{0})$ . In the difference quotient of $f_{x}$ the variable $y$ stays fixed at the value $y_{0}$ , so in the computation one differentiates with respect to $x$ treating $y$ as a constant, with the differentiation rules of one variable.

Example

$f (x, y) = x^{2} y^{3} + sin x$ . Differentiating with respect to $x$ with $y$ constant gives $f_{x} = 2 x y^{3} + cos x$ ; differentiating with respect to $y$ with $x$ constant gives $f_{y} = 3 x^{2} y^{2}$ , because $sin x$ is constant with respect to $y$ and its derivative is zero.

2. Gradient

If $f$ is partially differentiable at $(x_{0}, y_{0})$ , the gradient of $f$ at $(x_{0}, y_{0})$ is the vector of partial derivatives

\nabla f (x_{0}, y_{0}) = (f_{x} (x_{0}, y_{0}), f_{y} (x_{0}, y_{0})) .

For a function of $n$ variables the gradient has $n$ components. The gradient appears in the differential, in the formula for the directional derivative and in the search for critical points, so almost every exercise in this chapter begins with its computation. For $f (x, y) = x^{2} y^{3} + sin x$ we have $\nabla f (x, y) = (2 x y^{3} + cos x, 3 x^{2} y^{2})$ .

Differentiability

1. Differentiable function

$f : A \to R$ , $(x_{0}, y_{0})$ interior to $A$ . $f$ is differentiable at $(x_{0}, y_{0})$ if it is partially differentiable at $(x_{0}, y_{0})$ and

(h, k) \to (0, 0) lim \frac{f ( x _{0} + h , y _{0} + k ) - f ( x _{0} , y _{0} ) - f _{x} ( x _{0} , y _{0} ) h - f _{y} ( x _{0} , y _{0} ) k}{h ^{2} + k ^{2}} = 0.

The linear map $(h, k) \mapsto f_{x} (x_{0}, y_{0}) h + f_{y} (x_{0}, y_{0}) k$ is called the differential of $f$ at $(x_{0}, y_{0})$ . The condition says that the increment of $f$ is approximated by a linear function of the increments $h$ and $k$ , with an error that is $o (h^{2} + k^{2})$ :

f (x_{0} + h, y_{0} + k) = f (x_{0}, y_{0}) + f_{x} h + f_{y} k + o (h^{2} + k^{2}) .

In the exercises, differentiability at a point is checked by first computing the partial derivatives with the definition and then this limit.

Remark

In one variable, derivability and differentiability are equivalent. In two variables the existence of the partial derivatives is a weaker condition, since the partial derivatives control $f$ only along the two lines through $(x_{0}, y_{0})$ parallel to the axes, while the differentiability limit controls the increment along every way of approaching the point. For this reason, in several variables the central notion of differential calculus is differentiability.

2. Differentiability implies continuity

If $f$ is differentiable at $(x_{0}, y_{0})$ , then $f$ is continuous at $(x_{0}, y_{0})$ . The theorem is often used in the following form: if $f$ is not continuous at a point, then $f$ is not differentiable at that point.

Proof

By differentiability, as $(h, k) \to (0, 0)$ we have

f (x_{0} + h, y_{0} + k) - f (x_{0}, y_{0}) = f_{x} h + f_{y} k + o (h^{2} + k^{2}) .

The terms $f_{x} h$ and $f_{y} k$ tend to $0$ because they are products of a constant and an infinitesimal; the remainder tends to $0$ because it is an $o (h^{2} + k^{2})$ and $h^{2} + k^{2} \to 0$ . Hence $f (x_{0} + h, y_{0} + k) \to f (x_{0}, y_{0})$ , and this is the continuity of $f$ at $(x_{0}, y_{0})$ .

Warning

The existence of the partial derivatives at a point does not guarantee continuity at that point. The function

f (x, y) = ⎩ ⎨ ⎧ \frac{x y}{x ^{2} + y ^{2}} 0 (x, y) \neq = (0, 0) (x, y) = (0, 0)

is zero on the axes, so $f_{x} (0, 0) = f_{y} (0, 0) = 0$ and both partial derivatives exist. Along the line $y = x$ we have $f (t, t) = \frac{t ^{2}}{2 t ^{2}} = \frac{1}{2}$ for every $t \neq = 0$ , while along the $x$ -axis the function is zero, so the limit of $f$ at $(0, 0)$ does not exist and $f$ is not continuous at the origin. Since differentiability implies continuity, $f$ is not differentiable at $(0, 0)$ .

y = x

y = -x

f (x, y) = \frac{x y}{x ^{2} + y ^{2}}

equals

0

on the axes,

\frac{1}{2}

on the line

y = x

(in color) and

- \frac{1}{2}

on the line

y = - x

(dashed), so it has no limit at the origin

The theory continues with 6 more sections (tangent plane, the total differential theorem, directional derivatives and the gradient formula…), plus all worked examples, the solutions and the PDF.

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