The Taylor series is a way to express a function as an infinite sum of terms, each of which is derived from the function’s derivative value at a specific point .
Consider the power series . We have , , , and so on. Therefore,
Eq32b is called a Taylor series, which approximates a function in the neighbourhood of the point . When , the Taylor series is also known as a Maclaurin series:
The input value of in eq32b represents points in the domain of that are near . In other words, we can express as , where is a constant, is a variable, and represents a small change in that is scaled by . This alternate expression of is useful when dealing with multiplevariable functions.
For a multivariable function , the Taylor series about the point is
Question
How do we use vector notation to express the function ?
Answer
If we let , then we write in place of . In other words, we have , where . Consequently, can be regarded as

 a function of variables ,
 a function of a single point variable , or
 a function of a single vector variable .
To derive eq32c, let and . Consider the function
which implies that .
is a multivariable function, whose input is the vector , which varies with . In the domain of , is the vector representing a point where is expanded, and is a fixed vector that determines the direction of the displacement from the point . In other words, is not a variable of , but a parameter that we choose before plotting . The function is therefore a single variable function of , and its points are of the form .
The Maclaurin series expansion of when is
According to the multivariable chain rule, , where and . So,
The second derivative is
Using the multivariable chain rule again, we get
With reference to eq32d, with components . It follows that , and . Eq32g becomes
Furthermore, when , we have . So, eq32f and eq32h become
and
respectively.
Substituting eq32i and eq32j in eq32e and noting that and , we have eq32c. Finally, the Maclaurin series of a multivariable function is