Type of differential and integral calculus

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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