Last time, I sent a calculus formula, and a few classmates sent me a private message saying that I would talk about it in detail, and I was very embarrassed, and I willingly wrote the following explanation. Let's learn together
The content of the formula: Derivable must be continuous, continuous must be integrable, continuous must be bounded, integrable must be bounded, integrable is not necessarily continuous, continuous is not necessarily differentiable, can be differentiated, partial conductance must be differentiable, partial conductance existence is not necessarily continuous, continuous partial derivative exists, differentiable is not necessarily partial conductive continuous, second-order mixed partial conductance continuous partial derivative is equal, partial derivative is continuous and a bounded function is differentiable.
The differential and integral properties of the functions involved in the formula. The meaning of each mantra will be explained in detail below:
1.Derivative must be continuous: If a function is derivable at a certain point, then it must be continuous at that point. This is because derivability requires that both the left and right derivatives of the function exist and are equal at that point, whereas continuity requires that the limit value of the function exist at that point and is equal to the value of the function.
2.Continuous must be integrable: If a function is continuous within an interval, then it is integrable within that interval. This is one of the basic conditions for Riemann points.
3.Continuity must be bounded: If a function is continuous within an interval, then it is bounded within that interval. This is because the image of a continuous function will not have an "infinitely high" or "infinitely low" part, and can therefore be confined to a finite interval.
4.The integrable must be bounded: if a function is integrable within an interval, then it is bounded within that interval. This is another essential condition for Riemann points.
5.The integrability is not necessarily continuous: integrability does not require that the function be continuous at every point, for example, the Riemann sum function in a Riemann integral may be discontinuous at the split point.
6.Continuity is not necessarily differentiable: Continuity also does not guarantee that the function is differentiable at a certain point, for example, the left and right derivatives of a function at a point may not be equal, and the function is not differentiable at that point.
7.Differentiability must be continuous: If a function is differentiable at a certain point, then it must be continuous at that point, because differentiability contains the requirement of continuity.
8.Partial derivative continuous must be differentiable: If the partial derivative of a function exists and is continuous at a point, then the function is differentiable at that point. This is an important conclusion in multivariate calculus.
9.The existence of partial derivatives is not necessarily continuous: The existence of a partial derivative of a function at a point does not mean that the function is continuous at that point. For example, a partial derivative of a function may exist at a point, but the limit of the function at that point may not be equal to the value of the function.
10.Continuity does not necessarily have a partial derivative: Continuity also does not guarantee the existence of a partial derivative of a function at a certain point, for example, the change in the value of a function at a certain point may be different in all directions.
11.Differentiability is not necessarily partial derivative continuous: if a function is differentiable at a certain point, its partial derivative may exist but is not continuous.
12.Equal partial derivatives of second-order mixed partial derivatives: If the second-order mixed partial derivatives of a function at a point are continuous, then these partial derivatives are equal at that point. This is the symmetry of the second derivative.
13.A partial derivative is continuous and a bounded function is differentiable: If a function is continuous at a point with a partial derivative and another partial derivative is bounded, then the function is differentiable at that point.
These formulas summarize some of the basic properties and theorems of functional differentiation and integration, and are very helpful for understanding related concepts in advanced mathematics. Understanding and mastering these properties is important for solving specific problems in practical learning and application. Especially your last week of revision!