Functions of several variables · Definition

Smooth function

analysisdifferential geometry

Before you start you need to know

A smooth function can be differentiated as many times as we like, and it is the basic object of differential geometry.

Definition

A smooth function is a function between spaces with differentiable structure that, written in local coordinates, has derivatives of all orders.

\[f\in C^{\infty}(M,N)\]

Also called: of class C^∞

Properties: sums, products and compositions of smooth functions are smooth, and polynomials and the exponential on \(\mathbb{R}\) are \(C^{\infty}\).

Guide example: \(f(x)=x^{5}\) has continuous derivatives of every order and its sixth derivative is zero, so \(f\) is \(C^{\infty}\).

Typical error: thinking one derivative is enough. \(x|x|\) is \(C^{1}\), but its second derivative \(2\,\mathrm{sgn}(x)\) jumps at \(0\).

Think about it

What is the difference between being \(C^{1}\) and \(C^{\infty}\)?

\(C^{\infty}\) requires continuous derivatives of all orders, while \(C^{1}\) only of the first order.

Exercises 10

  1. What does it mean for \(f\) to be of class \(C^{\infty}\)?
  2. Which of these functions from \(\mathbb{R}\) to \(\mathbb{R}\) is smooth?
  3. Which symbol denotes the set of smooth functions from \(M\) to \(N\)?
  4. What class is \(f(x)=x^{5}\)?
  5. Let \(f(x)=x|x|\). What class is \(f\)?
  6. Let \(f(x)=x^{2}\) if \(x\ge 0\) and \(f(x)=0\) if \(x<0\). Is \(f\) smooth?
  7. If \(f\) and \(g\) from \(\mathbb{R}\) to \(\mathbb{R}\) are smooth, what can be said about \(g\circ f\)?
  8. Let \(f(x)=x^{2}\sin(1/x)\) for \(x\ne 0\) and \(f(0)=0\). Which statement is correct?
  9. What is a diffeomorphism between two smooth manifolds?
  10. Let \(f(x,y)=x^{2}y+\sin y\). What is \(\dfrac{\partial^{2} f}{\partial x\,\partial y}\)?

References

  • John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer, 2013.