A quick trip through some essential concepts and language.

It follows that, for each

A function is continuous if its graph can be plotted without breaks. X and Y are not necessarily sets of scalars. |

In spaces with curvature, coordinates are not vectors; a curve is not, in general, a vector valued function. A curve may be expressed in terms of coordinate functions of a parameter,

Sometimes the notation is used for

Sometimes

1*f* = *f* 1 = *f*.

For a function If the functions

(*g f* )^{−1} = *f*^{ −1}*g*^{−1}.

Clearly, the composition of any function For example, bras are functionals on a vector space of kets.

The vector spaces used to define an operator can be, but are not necessarily, the same. Since scalars are a one dimensional vector space, functionals and functions can both be regarded as operators. For the operators

Thus functions, functionals and operators can be treated as vectors. In physics, we only require answers at a finite range and resolution. It is sufficient to regard them as

Let

Thus, to specify a linear operator, we only have to specify its action on a basis. Converting to coordinate notation, and using the summation convention,

If H

In general the order in which functions and operators act affects the result; [

[*aA + bB*, *C*] = *a*[*A*, *C*] + *b*[*B*, *C*].

[*C*, *aA + bB*] = *a*[*C*, *A*] + *b*[*C*, *B*].

[

[*A*, *B*] = −[*B*, *A*].

[[*A*, *B*], *C*] + [[*B*, *C*], *A*] + [[*C*, *A*], *B*] = 0.

Thus, the inner product between and is the same as the inner product between and . Clearly, for any operators

Since reversing an inner product is complex conjugation,

So, the matrix form of

Inner products formed with Hermitian operators are real,

This property gives Hermitian operators a central importance in the description of observable quantities in quantum theory.

For a unitary operator

Thus a unitary operator preserves the inner product. In particular, vector magnitudes are preserved by unitary operators (unitary operators are isometries). A unitary transformation is a transformation described by a unitary operator.

A derivative, when it exists, is an approximation to the slope, or gradient, of the tangent to a graph, using a value of dx so small that making it any smaller makes no practical difference to the result. The differential operator maps a function to its derivative. |

It is straightforward to show that D

where

A second order (partial) derivative is found by (partial) differentiation of a (partial) derivative. An

It is straightforward to show that ∂

where the summation convention has been used (proof). Since this is an invariant and is a contravariant vector,

For a coordinate transformation,

So, the transformation matrix can be written using partial derivatives,

Functions, Operators, and Derivatives ↑ Introduction to Tensors →

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