I would like to explain to you additional types of *differential and integral calculus* for mathematics in university. To explain them, there are needed strict
and exact conditions for the function, and yet I will only use the expression
*smooth functions* so that senior-high-school students can also understand my explanation.

[Partial differentiation]

A function of two variables will generate a surface in three-dimensional space. Differentiating the function with respect to or is termed *partial differentiation* and described as:

Naturally, there is also *partial differential equation*" like *differential equation*.

[Implicit Function]

If there is a relation , then there is also a function satisfying and is termed implicit function. Then, the following equation stands:

[Multiple Integral]

Integrating with respect to a domain in a plain is termed *multiple integral* and described as:

The domain means various shapes including rectangle, circle, triangle, a domain bounded by two functions and .

For example, let is generated by satisfying : then

represents a volume of the domain bounded by the lines in the figure below.

[Curvilinear Integral]

Integrating Surface in three-dimensional space along Smooth curve is termed curvilinear integral and described as:

It means solving the area of the domain as illustrated in the figure below.

[Complex Integral]

is assumed as a function from a complex number to a complex number . Complex integral is like curvilinear integral, integrating on a curve assuming Curve in plain. It is described as:

Complex integral has relatively different characters from real integral.

[Lebesgue Integral]

This means integrating under the consideration that is a set. *Lebesgue Integral* defines integral without a concept of differentiation, while a condition
of *differentiable* is needed for *Riemann Integral* as above-mentioned up to *Complex Integral*. For example, Lebesgue Integral is used to strictly treat of probability
that will never permit *smooth* treatment.

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