And so on and so forth to a higher order term.

Then without moving your point from 0 you can predict all

the values that the function is that function of x.

So how is that possible?

It is possible because derivatives here shows the relationship

between neighboring points.

So knowing the relationship between neighboring points you can extend that

relationship to cover all space all space in broad.

So that's the power of derivatives.

And most of the time as you may learn the higher order term or

contribute less and less to the whole function.

So knowing the first order or second order might be sufficient.

So in this class, we're going to only cover the first order derivatives and

second order derivatives to understand the characteristics of our vector field.

So that's why we will start with the first order derivative.

So how shall we take the derivative of temperature with respect to position will

be the related question here. >> Okay.

>> And one guess might be,

how about round T over round x, round T over round x?

Note that this is partial derivative and it's not the direct

derivative, and Melody, if we do a round T over round x,

will that it be a vector field? >> No.

>> Or a scalar field?

>> No.

>> And why is that the case?

>> Because if you shift the coordinate

system you won't get a similar value- >> Exactly.

>> For your equation.

>> So the important thing here is scalar

and vectors are inherent upon the choice of the coordinates.

So we will learn more about how we can prove one field is a scalar or a vector

field using the operation that we just learned, namely the dot product operation.

So it is true only if, when we rotate the coordinate system the components

of the vector transform among themselves in the correct way.

So this will be another way to prove whether it is a vector field.

So ask a question who's answer is independent of the coordinate system and

try to express the answer in an invariant form.

So here is an example, if S, which is a scalar,

= A.B and if A and B are vectors, S is a scalar.

Likewise, if A is a vector, S a scalar, and

there are 3 numbers, B1, B2 and B3 that satisfied

the relationship A x B1 + A Y B2 + A 2 B3 this is

the operation of dot product equals S then B1, B2,

B3 are the components Bx, By, Bz of some vector B.

So this is quite an abstract idea, so let's take a look at

some specific example to understand what this means.

So, in this example we are going to prove that round T over round x,

round T over round y, round T over round z.

Constitutes a vector, okay?

In order to do that, let's take a look at this picture.

So, think about a contingent coordinates, where you have x and y and z.

In perpendicular fashion and

imagine you have two points that are very close to each other, P1 and P2.

And connect those points with an arrow which

will be the relative position vector of P2 with respect to P1 and

think of this as a diagonal vector for

a box that is drawn as a dotted line here, okay?

In this case we already learned temperature is a scalar field, all right?

And let's think of the temperature at P1 and P2.

And also lets think about the temperature difference of delta T between those two

points.

Because T2 a scalar and T1 is scalar.

Subtraction of those two, delta T will be also scalar.

It's inherent upon the choice of the coordinates, right?

And when T1 nad T2 are temperatures that P1 and

P2 are separated by the small interval delta R which is the relative

position vector then we can use delta x, delta y, delta z.

And we also know that the position vector is a vector field, all right?

So using the equation that I asked you to memorize where

the change of a function can be described by linear

combination of the change along each axis, which is here.

Then using this formula and replacing f by T, temperature,

then you can understand the change in temperature will

be equal to round T over round x times delta x + round T

over round y times delta y + round T over round z delta z.

When these change becomes infinitesimally small, approaching zero, right?

Then, as you look into this equation, this is the operation

of dot product between the relative position vector and

the three numbers that I just wrote here, okay?

Will be dot product operations.

So, we know delta R is vector, we know delta T's scalar,

therefore, these three numbers constitute a vector.

So that's the end of the proof.

So this might be a very easy and convenient way to prove

that three numbers, whether three numbers constitute a vector or not, but

another way We will shortly discuss after