Mean Value Theorem - Mean Value Theorem in Several Variables

Mean Value Theorem in Several Variables

The mean value theorem in one variable generalizes to several variables by applying the theorem in one variable via parametrization. Let G be an open subset of Rn, and let f : G → R be a differentiable function. Fix points x, y ∈ G such that the interval x y lies in G, and define g(t) = f((1 − t)x + ty). Since g is a differentiable function in one variable, the mean value theorem gives:

for some c between 0 and 1. But since g(1) = f(y) and g(0) = f(x), computing g′(c) explicitly we have:

where ∇ denotes a gradient and · a dot product. Note that this is an exact analog of the theorem in one variable (in the case n = 1 this is the theorem in one variable). By the Schwarz inequality, the equation gives the estimate:

In particular, when the partial derivatives of f are bounded, f is Lipschitz continuous (and therefore uniformly continuous). Note that f is not assumed to be continuously differentiable nor continuous on the closure of G. However, in the above, we used the chain rule so the existence of ∇f would not be sufficient.

As an application of the above, we prove that f is constant if G is connected and every partial derivative of f is 0. Pick some point x₀ ∈ G, and let g(x) = f(x) − f(x₀). We want to show g(x) = 0 for every x ∈ G. For that, let E = {x ∈ G : g(x) = 0} . Then E is closed and nonempty. It is open too: for every x ∈ E,

for every y in some neighborhood of x. (Here, it is crucial that x and y are sufficiently close to each other.) Since G is connected, we conclude E = G.

Remark that all arguments in the above are made in a coordinate-free manner; hence, they actually generalize to the case when G is a subset of a Banach space.