Real Analysis/Differentiation in Rn
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We will first revise some important concepts of Linear Algebra that are of importance in Multivariate Analysis. The reader with no background in Linear Algebra is advised to refer the book Linear Algebra.
Vector Space
editA set is said to be a Vector Space over a field if and only if operations addition and scalar multiplication are defined over it so as to satisfy for all and
(i)Commutativity:
(ii)Associativity:
(iii)Identity:There exists such that
(iv)Inverse:There exists such that
(v):
(vi)
(vii)
Members of a vector space are called "Vectors" and those of the field are called "Scalars". , the set of all polynomials etc. are examples of vector spaces
A set of linearly independant vectors that spans the vector space is said to be a Basis for the vector space.
Linear Transformations
editLet be vector spaces.
Let
We say that is a Linear transformation if and only if for all ,
(i)
(ii)
As we will see, there are two major ways to define a 'derivative' of a multivariable function. We first present the seemingly more straightforward way of using "Partial Derivatives".
Directional and Partial Derivatives
editLet
Let
We say that is differentiable at with respect to vector if and only if there exists that satisfies
is said to be the derivative of at with respect to and is written as
When is a unit vector, the derivative is said to be a partial derivative. Here we will explicitly define partial derivatives and see some of their properties.
Let be a real multivariate function defined on an open subset of
- .
Then the partial derivative at some parameter with respect to the coordinate is defined as the following limit
- .
is said to be differentiable at this parameter if the difference is equivalent up to first order in h to a linear form L (of h), that is
The linear form L is then said to be the differential of at , and is written as or sometimes .
In this case, where is differentiable at , by linearity we can write
is said to be continuously differentiable if its differential is defined at any parameter in its domain, and if the differential is varying continuously relative to the parameter , that is if it coordinates (as a linear form) are varying continuously.
In case partial derivatives exists but is not differentiable, and sometimes not even continuous exempli gratia
(and ) we say that is separably differentiable.
Total Derivatives
editThe total derivative is important as it preserves some of the key properties of the single variable derivative, most notably the assertion differentiability implies continuity
Let
We say that is differentiable at if and only if there exists a linear transformation, , called the derivative or total derivative of at , such that
One should read as the linear transformation applied to the vector . Sometimes it is customary to write this as .
Theorem
editSuppose is an open set and is differentiable on A. Think of writing in components so . Then the partial derivatives exist, and the matrix representing the linear transformation with respect to the standard bases of and is given by the Jacobian Matrix:
evaluated at .
NOTE: This theorem requires the function to be differentiable to begin with. It is a common mistake to assume that if the partial derivatives exist then this would imply that the function is differentiable because we can construct the Jacobian matrix. This however is completely false. Which brings us to the next theorem:
Theorem
editSuppose is an open set and . Think of writing in components so . If exists and is continuous on for all and for all , then is differentiable on .
This theorem gives us a nice criteria for a function to be differentiable.