00:01
The vector field is said to be conservative when the curl of it is zero the curl of the vector field is a zero vector so let's verify whether this is a conservative field or not so del cross f will be ijk do by though x do by two y do by two z e power 6x cost 6y minus e power 6x sine 6y z component is 0.
00:46
So let's evaluate the determinant i into though by the y of 0 is 0 minus though by 2 z of e power 6x 9 6x 9 65 is also 0 minus j into though by the x of 0 minus though by 2 z of e power 6x cost 65 is also 0 plus k into though by the x of minus e power 6x sine 6y is minus 6 e power 6x sine 6 y minus 6 x x x sine 6 y minus 6 x x x x sine 6 y minus minus what is though by though by of this guy, e power 6x cost 6y, it is minus e power 6x sine 6y into 6 because differentiation of cost is negative sign and this minus minus becomes plus and this whole term gets cancelled so it is 0 so that means the curl of f is 0 vector so that means f is a conservative vector field f is a conservative vector field.
01:55
Now since f is a conservative vector field, we can always write f as del 5, del phi, where phi is a scalar value function, where phi is a scalar value function.
02:13
By definition, del phi is do phi by though x i cap plus do phi by though by j cap.
02:20
I am not writing k -cap because there is no z component.
02:25
Comparing though phi by though x is equal to, what is the component of f, e power 6x, cos 6y.
02:35
And though 5 by though y is equal to minus e power 6x, sine 6y.
02:45
Integrating partially with respect to x, so you get 5 is equal to integrating partially with respect to x...