Five new video tutorials and five new solved problems are up in the Projectile Motion section of the Dynamics course! They introduce the general method for solving these types of problems, and go through a few different cases of one and two dimensional motion.

Engineer4Free has recently been included in Feedspot's Top 60 list of Civil Engineering Blogs and websites! Very excited and honoured to make the cut among many other incredible sites. 

Check out the post below to browse some of the world's top online resources for Civil Engineers. 

The Complex Numbers section of the Linear Algebra Course is now complete. There are free tutorials that cover the concept introduction, some operations, and also how to express complex numbers in polar form.
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• A quick introduction to complex numbers
• How to add and subtract complex numbers
• How to multiply complex numbers
• How to divide complex numbers
• How to find the absolute value of a complex number
• How to write a complex number in polar form

I created 36 free Differential Equations tutorials that focus on first and second order differential equations. The course covers basic introductions, integrating factors, separable differential equations, Bernoulli differential equations,  exact differential equations, non-exact differential equations, Euler's method, characteristic equations, fundamental solution sets, wronskian, reduction of order, undetermined coefficients, and variation of parameters! There are tons of detailed explanations and worked examples to help you better understand and grasp Differential Equations. ​

15 new tutorials covering second order differential equations are now up! They cover a few important things about the characteristic equation, as well as reduction of order method, undetermined coefficients methods, and variation of parameters. 
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• Characteristic equation and general solution of a linear second order DE 
• Characteristic equation example #1: real distinct roots
• Characteristic equation example #2: real repeated roots
• Characteristic equation example #3: complex roots
• Fundamental solution set and wronskian
• Reduction of order method
• Reduction of order example
• Undetermined coefficients method
• Undetermined coefficients example #1
• Undetermined coefficients example #2
• Undetermined coefficients example #3
• Variation of parameters method
• Variation of parameters example #1
• Variation of parameters example #2
• Variation of parameters example #3

Wtf are eigenvalues and eigenvectors? I had that same thought and felt like I wasn't alone, so I went and figured it out and made a bunch of youtube tutorials about them. 

PS, did you know that "eigen" more or less means "my own" in Dutch? You'll understand the relevance once you go through the videos. 

Wtf are eigenvalues and eigenvectors 
Show that x is an eigenvector of A and find the corresponding eigenvalue
Show that λ is an eigenvalue of A and find a corresponding eigenvector
Find the characteristic polynomial of a matrix
Find the eigenvalues and eigenvectors of a 2x2 matrix
Find the eigenvalues and eigenvectors of a 3x3 matrix

9 new tutorials are up on matrix algebra. This is your last chance. After this, there is no turning back. You take the blue pill—the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill—you stay in Wonderland, and I show you how deep the rabbit hole goes. Remember: all I'm offering is the truth. Nothing more. 

• The linear algebra matrix introduced
• Matrix addition, scalar multiplication, and "subtraction"
• Matrix multiplication
• Matrix transpose
• Find the determinant of a 2x2 matrix
• Find the determinant of a 3x3 matrix using cofactor expansion
• Find the determinant of a 3x3 matrix the fast way
• Find the inverse of a 2x2 matrix
• Find the inverse of a 3x3 matrix using the Gauss-Jordan method

After a very long hiatus, a new section is up in the linear algebra course covering vector combinations and span. There is about a 5 year gap between the publishing dates of video #25 and #26, whoops. Oh well, I'm back at it again with linear algebra and hope to finish off the set this month. Here are the new videos:

• How to determine if one vector is a linear combination of a set of vectors 
• How to determine if one vector in the span of a set of vectors 
• Write a vector as a linear combination of a set of vectors 
• How to determine if a set of vectors are linearly dependent or independent 
• Write a dependence equation for a set of vectors 
• Does a set of vectors span R^n
• What is a subspace
• Find a basis for the span of a set of vectors (either a subspace or a vector space)
• Some thoughts on linear combinations and span​​