00:01
According to the question, we are given that f x, y which is equals to 2 x y, x square plus 4 of e power of y is a vector field.
00:22
Alright, so we are given in the question.
00:25
Now what we have to do is, we need to show that the vector field is conservative.
00:32
So firstly we need to know that what's the process of showing the given vector field should be conservative.
00:39
So you will say that if the curl of f which is the curl of the given vector field is equals to 0, then we say that it is conservative.
00:50
Okay, so you can write in the bracket for conservative nature.
00:55
Alright, so what you need to do is that, you need to find out the curl of f for this given vector field and after that you need to check that either it is equals to 0 or not.
01:07
So curl of f is defined by del cross f and we very well know that del is defined by i, the unit vector, del by del x.
01:22
Then we have j, the unit vector, del by del y.
01:26
And then finally after that we have the unit vector k along with del by del z.
01:32
Alright, so what we need to do is that, we need to find out the curl of f using determinant method.
01:38
So we will write del cross f is equal to, so we will be making two straight lines.
01:44
Okay, and the first one is not straight, so pardon me for that.
01:49
So over here we will write the first unit vector i, j, k followed by j and k respectively and then we need to write the coefficients of i, j and k for del which is del by del x, then we have del y, del y and then we have del by del z.
02:09
Alright, and then finally we need to put the coefficients for i, j and k for the vector field.
02:14
So for the coefficient of i we have to x, y.
02:18
Okay, this first one is the coefficient of i, second one is the combination of the second unit vector, it's x square plus of e power of y, that is with the unit vector j and we're not having any unit, i mean we're not having any variable or any function assigned with the unit vector k, so we'll simply put it as zero...