chain Rule of partial derivitives

I realize that i said that i want this blog to be allabout linear algebra but i feel that that limits this blog alot. If your up for a little advanced calculus please sit bak relax and enjoy my chain rule.

So if you remeber chain rule from your first year maths course this will be very similer and i hope that we dont get too tripped up. Before we actually begin that let’s first review the notation for the chain rule for functions of one variable.

The notation that’s probably familiar to most people is the following.

There is an alternate notation however that while probably not used much in Calculus I is more convenient at this point because it will match up with the notation that we are going to be using in this section.  Here it is.

Notice that the derivative  really does make sense here since if we were to plug in for x then y really would be a function of t.  One way to remember this form of the chain rule is to note that if we think of the two derivatives on the right side as fractions the dx’s will cancel to get the same derivative on both sides. 

Wow that was a mouth full and haven't even began yet. if your need 5 miniuts. are you good ? ok lets go. at the end of this post there will be a pictre of a chained potato, uts try to make it to the end for that potato it needs you.

Lets look at a sample problem,  

Where :

,  

and we try to compute

This case is analogous to the standard chain rule from Calculus I that we looked at above.  In this case we are going to compute an ordinary derivative since z really would be a function of t only if we were to substitute in for x and y.

The chain rule for this case is,

So, basically what we’re doing here is differentiating f with respect to each variable in it and then multiplying each of these by the derivative of that variable with respect to t.  The final step is to then add all this up.

ill leave finding examples to your discression. if you would like me to post practice problems ill be happy to just let me know in the discussion below. 

As promised your chained potato!!!

Vectors in Rn Pt1

the rescource page can be found here notes

lets start of with the most basic quantifiable aspect of linear algebra. The humble vector. so what is a vector???
to explain that lets begin with what a vector is not. a vector is not a certain game on android that gets you to run for hours. a vector is a quantity. Now there are more types of quantities than just the vector. If you were to count the number of steps you have walked today on your fit-bit, or the temperature of the weather in your area to know what you should wear today. those measurements are not vectors but scalers, and its actually a very smart name for them that we will come back to in a moment. if you were to take that measurement such as the number of steps you've taken and add another piece of information, your direction, you will have recorded a vector. and there you go by definition a vector is a quantity that posses both magnitude and direction. we usually express these things by arrows. you may see the notes for an example of how they look. Because we love computer science here as well lets code this quantity.

I've written this in Java and posted it on my resource page that can be found here. I have sectored the source code off so that as you complete sections in this text you may reference my code and build up your own.

lets move onto section 2 now vector operations. if you are following along in the code we will move onto the operations package, the vectors class. lets begin with vector addition. if you have say two vectors in R2 [0, 1] [1, 1]

if we want to add these we will take each individual component and add them
[0+1, 1+1] = [1, 2].

remember above here when we were talking about those simpler quantities called scalers here I will explain them in a little more detail. you simply take the scaler a and multiply each component of the vector S e.g.:
a[1,2,3,4] =
[1a,2a,3a,4a] =
[a,2a,3a,4a]

you see what i mean by its such a clever name the scaler simply scales the vector to a greater magnitude but it will always still point in the same direction as before it was put through scaler multiplication.

thanks for reading here is a vector potato

all about maths

This blog will be a place for me to post maths problems and most of all a place for me to post maths notes while i study for my own maths course. I hope that we will be able to learn together. If you have any requests or opinions to share please post them in the comments. lets get to maths our most loved and favourite subject of them all.

we will have 2 main focuses right now on this blog and that will be advanced calculus and linear algebra. if you have a particular inclination i will be seperating these 2 topics and the various others we will touch on.

Disclaimer. this is not a gr 1 -12 year maths website we talk about real maths and hopefully you'll enjoy it as much as the rest of us do.