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**Topics covered:** Final review

**Instructor:** Prof. Denis Auroux

Lecture 34: Final Review

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Lecture Notes - Week 14 Summary (PDF)

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OK, so anyway, let's get started.

So, the first unit of the class, so basically I'm going to go over the first half of the class today, and the second half of the class on Tuesday just because we have to start somewhere.

So, the first things that we learned about in this class were vectors, and how to do dot-product of vectors.

So, remember the formula that A dot B is the sum of ai times bi.

And, geometrically, it's length A times length B times the cosine of the angle between them.

And, in particular, we can use this to detect when two vectors are perpendicular. That's when their dot product is zero. And, we can use that to measure angles between vectors by solving for cosine in this.

Hopefully, at this point, this looks a lot easier than it used to a few months ago. So, hopefully at this point, everyone has this kind of formula memorized and has some reasonable understanding of that.

But, if you have any questions, now is the time.

No? Good.

Next we learned how to also do cross product of vectors in space -- -- and remember, we saw how to use that to find area of, say, a triangle or a parallelogram in space because the length of the cross product is equal to the area of a parallelogram formed by the vectors a and b.

And, we can also use that to find a vector perpendicular to two given vectors, A and B.

And so, in particular, that comes in handy when we are looking for the equation of a plane because we've seen -- So, the next topic would be equations of planes.

And, we've seen that when you put the equation of a plane in the form ax by cz = d, well, b, c> in there is actually the normal vector to the plane, or some normal vector to the plane.

That's the velocity of the moving point on the line.

Then you have two vectors given on the plane.

But, very often, that's not so useful.

And, let's say that this plane is maybe given to you.

Let's say c1x c2y plus d, whatever, something like that.

And, those directions would be probably the xy plane.

So, I would look at the xy coordinates.

OK, any other questions on that? No?

And, its length will be the speed.

That's the geometric interpretation.

So, just to provoke you, I'm going to write, again, that formula that was that v equals T hat ds dt.

But, blue has been abolished since then.

And, what's the length? Well, it's the speed.

That's another guy. That's another of these guys for the speed, OK?

And, we've also learned about acceleration, which is the derivative of velocity.

So, it's the second derivative of a position vector.

I mean, we've seen those on the first exam.

OK, so if you don't remember how to multiply matrices, please look at the notes on that again.

And, also you should remember how to invert a matrix.

The method doesn't work, doesn't make sense.

Otherwise, then the concept of inverse doesn't work.

And, if it's larger than 3x3, then we haven't seen that.

So, let's say that I have a 3x3 matrix.

What I will do is I will start by forming the matrix of minors.

And, I would be left with this 2x2 determinant.

I take this times that minus this times that.

I get a number that gives my first minor.

I swept the rows and the columns.

And then, I divide by the inverse.

It's a fairly straightforward method.

You just have to remember the steps.

But, of course, there's one condition, which is that the determinant of a matrix has to be nonzero.

So, in fact, we've seen that, oh, there is still one board left.

The determinant of a matrix real quick?

Its determinant will be obtained by doing an expansion with respect to, well, your favorite.

Right, when we do cross products, we are doing an expansion with respect to the first row.

That's a special case. OK, I mean, do you still want to see it in more details, or is that OK?

Yes? How do you tell the difference between infinitely many solutions or no solutions?

You know, they can all pass through the same axis.

Then, the last one is actually in front of that.

And, there's no triple intersection.

It's called the trivial solution.

It's the obvious one, if you want.

So, you know that, and why is that?

Well, I'll let you figure it out.

OK, so what was the second part of the class about?

So, for each x and y, I plot a point at height given with the value of the a function.

But, if you have a function of three variables, you can do the same kinds of manipulations.

OK, so that lets us understand rates of change.

So, the value doesn't change, OK?

So, critical points, remember, are the points where all the partial derivatives are zero.

So, then we've seen the method of Lagrange multipliers.

I mean, this is only in the case of independent variables.

Will we actually have to calculate?

Well, that depends on what the problem asks you.

It might ask you to just set up the equations, or it might ask you to solve them.

And then, we have the partial f, partial x with y held constant, which means y held constant.

X varies, and now we treat z as a dependent variable.

It varies with x and y according to whatever is needed so that this constraint keeps holding.

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