1.6.1 Single Variable Functions
22.214.171.124 - 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,
126.96.36.199 - 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’,
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.
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.
188.8.131.52 - 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
184.108.40.206.1 - 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.
220.127.116.11.2 - 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.
18.104.22.168.3 - 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.
22.214.171.124 - Second Order Differential Equations
• These equations have at least one second order derivative.
• In engineering we will encounter a number of forms,
126.96.36.199.1 - Linear Homogeneous
• These equations will have a standard form,
• An example of a solution is,
188.8.131.52.2 - 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,
184.108.40.206 - Higher Order Differential Equations
220.127.116.11 - Partial Differential Equations
• Partial difference equations become critical with many engineering applications involving flows, etc.