Essential Formulas and Theorems in Multivariable Calculus and Differential Equations

Posted on Dec 15, 2024 in Mathematics

Convergence of Sequences

Convergence Point

To resolve the limit:

Uniform Convergence

  • State the Supremum Criterion:

To check for uniform convergence, the characterization of the supremum criterion is used, which states that converges uniformly to in M if and only if:

  • Calculate

  • Calculate the maximum (supremum) that can be reached by the absolute value in the interval marked by
  • Check that the supremum tends to zero as

Power Series

Series of the form:

Convergence Study

  • Calculate the radius of convergence:

If the series is complicated, it can be derived for and the radius of convergence of the derivative can be calculated, since it has the same radius.

  • The series converges absolutely for . It does not converge outside of this range.
  • Study the endpoints (If r is )
    • Replace in the original series with one of the values
    • Check whether the series converges or not. A rapid method is the criterion of the remainder, which states that if , the series is divergent.

Power Series Expansion

Continuity of Multivariable Functions

Limits

  • Check if there is an indeterminacy.

Values that nullify the function, i.e., return it as indefinite.

  • Calculate iterated limits.
    where

    • If they match, go to Step 3.
    • If they mismatch, there is no limit.
  • Calculate directional limits.


    If all the directions match the value of the iterated limits, go to step 4.
    If at least one does not, then the limit does not exist.
  • Demonstrate the existence of the limit.
    • By the definition of a limit.
    • By changing to polar coordinates.
  • Replace:
  • Make sure:

  • And it must fulfill:

Differentiability

Notation

  • Partial Derivative:
  • Directional Derivative:
  • Differential:

Calculating the Differential of a Function

If all partial derivatives exist and are continuous, we can say that is continuous and differentiable. If they are not continuous, we cannot say anything about continuity or differentiability.

If the function presents areas of conflict or is piecewise, you can use the definition of a directional derivative to demonstrate its continuity:

After verifying that the function is differentiable, the differential can be calculated as:

Relations

Multiplying matrices: multiply each row by each column and add:

Jacobian Matrix

The Jacobian matrix is defined by as:

Tangent Plane to a Function of Two Variables

Given a function , its tangent plane is of the form:

Setting the equation equal to zero yields the equation of the tangent plane.

Calculating Extrema of Multivariable Functions

In an Open Set

  • Find the critical points by solving the system:

  • Build the Hessian matrix of :

  • For solutions , verify that they are in the set, and if so, calculate the determinant of the Hessian matrix evaluated for . Then:
    • If the determinant is greater than zero and , then it is a minimum. If it is less than zero, it is a maximum.
    • If the determinant is less than zero, it is a saddle point.
    • If the determinant is zero, no information is obtained.

In an Empty Set (Lagrange Multipliers)

Given a function and a constraint (constraint equation)

  • Construct the function:

  • Solve the system:

  • If there is more than one critical point, simply compare the value of the function at each pair . Otherwise, it is necessary to calculate the discriminant of the form:

  • If , it will be a minimum if , and if not, we cannot determine its nature.

In a Compact Set (Closed and Bounded)

  • Find critical points in the interior using the method for open sets.
  • Define the boundaries and seek critical points on them. It may be necessary to use Lagrange multipliers, since the boundaries are empty sets.
  • Study and compare the value of for all critical points and vertices of the set.

Implicit Function Theorem

Given the function , a neighborhood, and , z may be solved for as an implicit function and if it satisfies:

  • Partial derivatives exist and are continuous
  • If the function has two coordinate functions, one finds that

Therefore, we can state the implicit function theorem:

We know there are open sets such that and and a function with the following properties:

Inverse Function Theorem

Given the function and

  • Check
  • State the inverse function theorem:

F is invertible in a neighborhood of and satisfies:

Taylor Polynomial for Multivariable Functions

Given the function , an order , and

  • Calculate:

Multiple Integration

These are integrals of the form:

Rectangular Region

When the region to be integrated is of the form , then the result is:

Basic Regions

When the region to be integrated is bounded, non-empty, and its boundary can be expressed with functions of a real variable, then by Fubini’s theorem:

Where

In many cases, a change of variables will be required to perform the integral. For example, when the region is a circle, it is convenient to use polar coordinates to solve the integral.

Variable Changes in a Multiple Integral

To change the type of coordinates in a multiple integral, apply the formula:

In polar coordinates, the Jacobian of the transformation is always .

Curvilinear Integration

These are integrals of the form:

By Parameterization

If the curve is expressed by a region, it can be configured to form a function and integrate along the curve in the form:

Where If has only one coordinate, then

If the curve is a semicircle, it can be expressed as:

By Green’s Theorem

Given a function and a bounded open set defining either and whose boundary ( ) is a differentiable simple closed curve, we integrate:

That is a multiple integral of the basic compound

Divergence Theorem

Given a bounded open set D whose boundary ( ) is a simple closed and differentiable curve, it satisfies:

  • Calculate the divergence of F:

  • Calculate the multiple integral of the basic region:

Checking if a Field is Conservative

Given a field , it is conservative if:

If there exists such that , then is called the potential function of .

Differential Equations

Separable Variable Equations

  • Leave the dependent values on one side of the equation and the independent values on the other.
  • Integrate both sides and solve for .

Homogeneous Equations

Of the form

  • Check that and are homogeneous of the same degree:

  • Make the change . Thus, the equation becomes a separable variable equation.

Linear Equations

Can be presented as:

This is called the homogeneous linear equation, and the equation is of separable variables. The solution is:

This is solved by the method of variation of constants. The solution is:

Method of Variation of Constants

  • Find the solution to the homogeneous equation (setting equal to zero). It will be of the form:

  • The constant of the homogeneous solution becomes a function.

  • Calculate
  • Substitute into the original equation ( ) the values obtained in the previous steps. If everything is correct, it should be possible to simplify and remove terms with in this step.
  • Solve for and replace its value in the equation from step 2.

Exact Equations

Of the form:

  • Check that it is exact if it satisfies:

  • Find satisfying and . This involves integrating M or N and finding . For example:
    • The solution will be

Integrating Factor

Equations that do not qualify as exact but can become exact by using an integrating factor such that:

Assuming an integrating factor that depends only on x:

  • Solve
  • Apply the factor and solve the resulting exact equation.

Some factors that can be tried: