## Local Homeo(Diffeo)morphisms to Global Homeo(Diffeo)morphisms

**Point of Post: **In this post we discuss an important consequence of the inverse function theorem which relates local diffeomorphisms to global diffeomorphisms.

## The Geometry of the Derivative for Real Valued Mappings (Pt. I)

**Point of Post: **In this post I’d like to discuss some of the geometric aspects of what the total and partial derivatives mean including the idea of approximating lines and tangent planes.

*Motivation*

As usual in math it’s helpful to have a picture to backup the ideas. In this post we discuss what it geometrically looks like when a mapping is differentiable at a point in terms of tangent planes. This of course generalize the notion that a mapping is differentiable at a point if it has a tangent line there.

## Functions of Class C^k

**Point of Post: **In this post we define the notion of classes of differentiability and discuss what the membership in some of these classes implies about the function.

*Motivation*

Since differentiability and related notions are our focus for right now, it seems prudent that we should define some kind of notation which shortens saying things like “ has partial derivatives of all types of order “, ” has partial derivatives of all types of order and each partial derivative is continuous”, or ” has partial derivatives of all orders and all types”. This is taken up by the notion of classes which, put simply tells you how well-behaved the function is.

## Relationship Between the Notions of Directional and Total Derivatives (Pt.II)

**Point of Post: **This is a continuation of this post.

## Relationship Between the Notions of Directional and Total Derivatives (Pt.I)

**Point of Post: **In this post we show the relationship between total and directional derivatives, and in doing so finally find an explicit formula for the total derivative in terms of the partial derivatives.

*Motivation*

So the question remains how the total derivative, which we said was a measure in-all-directions of local change and approximation to a function, and directional derivatives which we said was some kind of measure of change in a specified direction that ignored all others. Some things seem intuitively obvious. For example, one feels that functions which have total derivatives at a certain point should morally be obligated to possess directional derivatives in all directions. Moreover, it seems not too bizarre that the total derivative should be able to be expressed, in some way, by some combination of directional derivatives. I mean, it makes sense that the change in an arbitrary direction should have something to do with the way it’s “component” directions act. In fact, we shall prove both of these things–namely that total differentiability implies the existence of directional derivatives in all directions, and that the Jacobian can be expressed entirely in terms of partial derivatives. The surprising thing we shall show is that a fairly strong converse holds–namely that if all the partial derivatives are ‘nice’ (in a precise sense to define soon) then we are guaranteed total differentiability

## Directional Derivatives and Partial Derivatives

**Point of Post: **In this post we discuss the notions of directional derivatives and partial derivatives

*Motivation*

Roughly what the total derivative does is describe conditions when a function can be locally approximated very well (sublinearly) well by an affine transformation. Indeed, suppose that is differentiable at . By definition the limit for any we may choose such that implies . Note that that we see from this that ‘locally’ here means in all possible directions (as soon as is within the open ball the above inequality applies). Sometimes though we are only interested in the approximation, or notion of change in a particular direction. Thus arises the directional derivative which, roughly put, measures the rate of change of a function at a point towards a vector . The idea is simple, namely one takes, along the line from to , successive differences of the value and and let tend to zero. When the vector is one of the elements of the canonical basis we get the partial derivatives which, as we shall see, are of huge importance in multivariable differential analysis.

## Further Properties of the Total Derivative (Pt. II)

**Point of Post: **This is a continuation of this post.