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1.6.1 Single Variable Functions

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1.6.1.1 - Differentiation

• The basic principles of differentiation are,

• Differentiation rules specific to basic trigonometry and logarithm functions

• L’Hospital’s rule can be used when evaluating limits that go to infinity.

• Some techniques used for finding derivatives are,

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1.6.1.2 - Integration

• Some basic properties of integrals include,

• Some of the trigonometric integrals are,

• Some other integrals of use that are basically functions of x are,

• Integrals using the natural logarithm base ‘e’,

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1.6.2 Vector Calculus

• When dealing with large and/or time varying objects or phenomenon we must be able to describe the state at locations, and as a whole. To do this vectors are a very useful tool.

• Consider a basic function and how it may be represented with partial derivatives.

• Gauss’s or Green’s or divergence theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

• Stoke’s theorem is given below. Both sides give the flux across a surface, or out of a volume. This is very useful for dealing with magnetic fields.

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1.6.3 Differential Equations

• Solving differential equations is not very challenging, but there are a number of forms that need to be remembered.

• Another complication that often occurs is that the solution of the equations may vary depending upon boundary or initial conditions. An example of this is a mass spring combination. If they are initially at rest then they will stay at rest, but if there is some disturbance, then they will oscillate indefinitely.

• We can judge the order of these equations by the highest order derivative in the equation.

• Note: These equations are typically shown with derivatives only, when integrals occur they are typically eliminated by taking derivatives of the entire equation.

• Some of the terms used when describing differential equations are,

ordinary differential equations - if all the derivatives are of a single variable. In the example below ’x’ is the variable with derivatives.

first order differential equations - have only first order derivatives,

second order differential equations - have at least on second derivative,

higher order differential equations - have at least one derivative that is higher than second order.

partial differential equations - these equations have partial derivatives

• Note: when solving these equations it is common to hit blocks. In these cases backtrack and try another approach.

• linearity of a differential equation is determined by looking at the dependant variables in the equation. The equation is linear if they appear with an exponent other than 1.

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1.6.3.1 - First Order Differential Equations

• These systems tend to have a relaxed or passive nature in real applications.

• Examples of these equations are given below,

• Typical methods for solving these equations include,

guessing then testing

separation

homogeneous

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1.6.3.1.1 - Guessing

• In this technique we guess at a function that will satisfy the equation, and test it to see if it works.

• The previous example showed a general solution (i.e., the value of ’C’ was not found). We can also find a particular solution.

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1.6.3.1.2 - Separable Equations

• In a separable equation the differential can be split so that it is on both sides of the equation. We then integrate to get the solution. This typically means there is only a single derivative term.

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1.6.3.1.3 - Homogeneous Equations and Substitution

• These techniques depend upon finding some combination of the variables in the equation that can be replaced with another variable to simplify the equation. This technique requires a bit of guessing about what to substitute for, and when it is to be applied.

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1.6.3.2 - Second Order Differential Equations

• These equations have at least one second order derivative.

• In engineering we will encounter a number of forms,

- homogeneous

- nonhomogeneous

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1.6.3.2.1 - Linear Homogeneous

• These equations will have a standard form,

• An example of a solution is,

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1.6.3.2.2 - Nonhomogeneous Linear Equations

• These equations have the general form,

• to solve these equations we need to find the homogeneous and particular solutions and then add the two solutions.

• Consider the example below,

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1.6.3.3 - Higher Order Differential Equations

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1.6.3.4 - Partial Differential Equations

• Partial difference equations become critical with many engineering applications involving flows, etc.