00:02
In this problem, we are given two temperature distributions, t1 and t2, over a flat sheet.
00:08
And we want to demonstrate in question a that both of these temperature distributions obey the laplace equation.
00:17
So let's start with our first temperature distribution, t1, that is equal to e to the minus 2y times the cos of 2x.
00:27
So let's take t1 and differentiate partially with respect to x, assuming that y is constant.
00:33
And we obtain minus 2 times e to the minus 2y times the sin of 2x.
00:43
Now let's take this expression again and differentiate with respect to x.
00:49
And we find minus 4 times e to the minus 2y times the cos of 2x.
01:00
Going back to t1, let's differentiate this now partially with respect to y.
01:08
And we find minus 2 times e to the minus 2y times the cos of 2x.
01:14
And differentiating this expression again with respect to y, we obtain 4 times e to the minus 2y times the cos of 2x.
01:27
And now what i want to show is that the sum of these two last second order partial derivatives is equal to zero.
01:36
In fact, we see that the partial derivative of t1 with respect to x, the second degree, plus the partial derivative of t1 with respect to y, the second degree, is in fact equal to zero.
01:53
Thus demonstrating that t1 is a solution to laplace's equation.
01:59
Now consider t2 equal to the ln of the square root of x squared plus y squared.
02:16
Before we go any further, let's simplify our expression.
02:20
And we find 1 half times the ln of x squared plus y squared.
02:29
So let's take t2 and differentiate partially with respect to x.
02:33
And we obtain x over 1, excuse me, x over x squared plus y squared.
02:56
Now let's differentiate with respect to x again.
03:00
Now we're going to have to utilize quotient rule.
03:04
And we find x squared plus y squared minus x times 2x, so 2x squared.
03:21
Divided by x squared plus y squared squared.
03:31
Let's go back to t2 and differentiate partially with respect to y this time.
03:39
We find y over x squared plus y squared.
03:47
And differentiating this expression with respect to y again.
03:53
We find x squared plus y squared minus 2y squared over x squared plus y squared squared.
04:13
And now again i want to show that the sum of these last two second order partial derivatives is equal to zero.
04:21
And so summing these two, we find we obtain x squared plus y squared.
05:00
Oh i forgot, here this is division by 2, so there's no factor of 2 here.
05:09
So we obtain x squared plus y squared minus x squared minus y squared over x squared plus y squared squared...