Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 10 (Transcript)

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OK. So let me look at ...

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[...]

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Let me now look at this question.

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The question I mentioned er ...

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If you start at some point x zero, y zero on the map ...

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... and you want to go and make a step ...

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... in a certain direction ...

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... and your direction is not in direction of x or y -- some direction, right?

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So then this step can be denoted as a vector.

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Right? And what kind of vector?

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Well the vector in the coordinates x and y ...

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... is going to be ... is going to mean ...

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The x-coordinate will be change of x.

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The y-coordinate will be change of y.

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We can safely call those dx and dy or delta x and delta y.

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Those are the same things.

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So you apply that vector to the point x zero, y zero.

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And you look at what happens there.

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So what does it mean?

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You look at what happens. You look at the change of the values f of ...

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... x plus dx, y plus dy minus f of ...

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... x zero, y zero ...

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[...] x zero and y zero there.

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So the difference of values at these two points ...

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... related to the ...

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What?

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... to the magnitude of that vector.

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To the length of your step as you measure it on a map.

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Magnitude of dx, dy.

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That's going to be the approximately the rate of change in that direction.

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And er ...

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The good thing about this quantity is that it can be ...

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... easily expressed using that ...

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... formula for differentials.

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So as we replace the difference of values of f with df.

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What is it that we see.

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You see that df is equal to ...

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... f_x dx plus f_y dy ...

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... all that is related to the magnitude of dx, dy.

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And then [...] momentarily recognize algebraically the numerator as ...

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... dot product, yes!

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Isn't it a dot product?

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So how do you put these together?

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How do you like them to be together?

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Making two vectors.

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And dot product within them.

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Student: [...].

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You put partials together of course.

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And you put dx, dy together.

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And then dividing by the magnitude of dx, dy ...

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... makes perfect sense.

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Because then the dot product with that vector divided by the magnitude of that vector ...

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... is ...

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... component ...

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... of the vector f_x, f_y of the vector made by partial derivatives ...

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... in direction of dx, dy.

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Is that any special direction?

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Well that's exactly the direction that we follow, right?

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So if you want to go from that point in that direction ...

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... then this component ...

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... tells you the rate of change in that direction.

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And ...

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That not only gives you a computable value.

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It tells you, hints you ...

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... that this vector has some importance.

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If you take ... well you never did this, you never put ...

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... partial derivatives of a function together as a vector.

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And now we do.

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So for given function f of x, y ...

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... if you take partial derivatives ...

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... and you put them together and make a vector ...

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... and the function of ...

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The function is of two variables.

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You have two partial derivatives.

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It is going to be a two dimensional vector.

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Then that's called the gradient vector.

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And that is actually considered as the derivative ...

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... of f.

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Why do you think it's ...

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... it should be considered as derivative? What should be a hint?

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Why is that the derivative?

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[...] as a question.

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I'll tell you tomorrow.

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