(1) interior point: If the point has a domain, it is called the interior point of.

(2) Outliers: If a domain exists at this point, it is called outliers.

(3) Boundary point: If any domain of a point contains both attribution points and non-attribution points, it is called a boundary point.

All the boundary points of are called the boundary of, and are recorded as.

The inner point of must belong to, the outer point of must not belong to, and the boundary point of may or may not belong to.

Gathering point: for any given point, if there is always a midpoint in the centripetal field of a point, it is called gathering point.

Open set: If all points of a point set are interior points, it is called open set.

Closed set: If the boundary of a point set is called a closed set.

Connected set: If any two points of a point set can be connected by a broken line, and all points on the broken line belong to it, it is called a connected set.

Region (or open region): Connected open sets are called regions.

Closed area: A point set formed by an open area and its boundaries is called a closed area.

Bounded set: for a plane point set, if there is a positive number, which is the origin of coordinates, it is called a bounded set.

Unbounded set: If a set is unbounded, it is called unbounded.

Let the domain of binary function be the aggregation point. If there is a constant, for any given positive number, there is always a positive number, which makes all points effective, then this constant is called the limit of the function, and it is recorded as

Let the domain of a binary function be the aggregation point of yes, and if it is, the function is said to be continuous at this point.

Let a function be defined on the graph, and every point in it is the aggregation point of the domain of the function. If the function is continuous at every point, it is said that the function is continuous in the world, or that the function is continuous in the world.

Let the domain of a function be an aggregation point, and if the function is discontinuous at this point, it is called a discontinuous point of the function.

All multivariate elementary functions (multivariate elementary functions refer to multivariate functions that can be expressed by a formula) are continuous in their defined areas, and the so-called defined areas refer to areas contained in defined areas or closed areas.

Let a function be defined in a domain of a point. When it is fixed at and there is an increment at that point, the corresponding function has an increment. If it exists, this limit is called the partial derivative of the function at this point, and it is recorded as

Similarly, the partial derivative of a function at a point is defined as

record

If a univariate function has a derivative at a certain point, it must be continuous at that point, but for multivariate functions, even if all partial derivatives exist at a certain point, there is no guarantee that the function must be continuous at that point, because the existence of each partial derivative can only guarantee that the function value will tend when the point tends in the direction parallel to the coordinate axis, but not when the point tends in any way.

Let a function have a partial derivative in the region.

Then, if the partial derivatives of these two functions also exist, it is called the second-order partial derivative of the function:

Theorem? If two second-order mixed partial derivatives of a function are continuous in the region, then the two second-order mixed partial derivatives must be equal in the region. That is to say, under the continuous condition, the second-order mixed partial derivative has nothing to do with the order of derivative.

Let a function be defined in a domain of a point, if the function is in the full increment of the point.

Can be expressed as

When it is not dependent on but only related to, a function is said to be differentiable at this point, which is called the total differential of the function at this point, and is recorded as

A function is said to be internally differentiable if it can be differentiated at every point in the region.

The existence of partial derivative of multivariate function at a certain point does not guarantee the continuity of function at that point; However, if a function is differentiable at a certain point, it must be continuous at that point.

Theorem 1 If the function is differentiable at one point, the partial derivative of the function at one point must exist, and the total differential of the function at one point is

Theorem 2? If the partial derivative of a function is continuous at a certain point, then the function can be derived at that point.

Suppose a function has a continuous partial derivative, then it has a total differential.

If it is also an intermediate variable, that is, and the two functions also have continuous partial derivatives, the total differential of the composite function is

It can be obtained from the formula in 4.2.

Theorem 1 If the sum of functions is differentiable at this point and the function has continuous partial derivatives at the corresponding point, then the composite function is differentiable and exists at this point.

This is called total differential.

Theorem 2? If the sum of functions has partial derivatives of pairs and pairs at points, and the function has continuous partial derivatives at corresponding points, then both partial derivatives of composite functions at points exist and exist.

Theorem 3? If a function has partial derivatives of pairs and pairs at one point, a function is derivable at one point, and a function has continuous partial derivatives at the corresponding point, then both partial derivatives of a composite function exist and exist at one point.

No matter to whom the derivative is derived and how many derivatives are obtained, the new function after derivative still has the same compound structure as the original function.

Implicit function 1 existence theorem? If a function has a continuous partial derivative in a certain domain at a certain point, then the equation can always uniquely determine a continuous function with a continuous derivative in a certain domain at a certain point, which satisfies the conditions and has

This formula will be brought into the equation and then derived from both sides of the equation.

Implicit function existence theorem 2? If a function has a continuous partial derivative in a certain domain at a certain point, then the equation can always uniquely determine a continuous function with a continuous derivative in a certain domain at a certain point, which satisfies the conditions and has

In the same way, bring it into the equation, and then take derivatives on both sides of the equation.

Implicit function existence theorem 3? Let a function have a continuous partial derivative for each variable in a certain domain at a point, and the partial derivative is the determinant of the function.

If this point is not equal to zero, the equations can always uniquely determine a continuous function with continuous derivatives in a certain domain at this point. They meet the requirements and

Derive both sides of the equation, and then solve the equation.

Let the parameter equation of the space curve be

Assume that all three functions are differentiable in the upper bound, and the three derivatives are not zero at the same time.

The parameter corresponding to a point is, remember, then the vector is the tangent vector of the curve at that point, and then the tangent equation of the curve at that point is

The plane passing through this point and perpendicular to the tangent is called the normal plane of the curve at this point, that is, the plane passing through this point and taking it as the normal vector. Therefore, the equation of the normal plane is

Let the surface be given by the equation and be a point on the surface, and let the partial derivative of the function be continuous at this point and zero at different times. The plane formed by the tangents of all curves passing through this point on the surface is called the tangent plane of the surface at this point, and its equation is

A straight line passing through a point and perpendicular to the tangent plane is called the normal of the point surface, and its equation is

The vector perpendicular to the tangent plane of the surface is called the normal vector of the surface. Vector

Is the normal vector of a surface at a certain point.