1. ## Vector field

1) How can I know if the field is gradient?

2) How can I know if the fiels is conservative ?

3) How can I find the function potential of vector field ?

2. Originally Posted by Apprentice123
1) How can I know if the field is gradient?

2) How can I know if the fiels is conservative ?

3) How can I find the function potential of vector field ?
1) If ${\bf F} = \nabla f$ for some $f.$
2) If $\nabla \times {\bf F} = 0$
3) Set $\nabla f = {\bf F}$ and solve for $f$.

3. if I have $F(x,y) = (e^xcosy)i + (-e^xseny)j$

what will be: $\nabla$ and $f$

4. Originally Posted by Apprentice123
if I have $F(x,y) = (e^xcosy)i + (-e^xseny)j$

what will be: $\nabla$ and $f$
Since you only have a two dimensional vector field, say

$
{\bf F} =
$

then this would be conservative if $P_y = Q_x$ and if $P = e^x \cos y$ and $Q = - e^x \sin y$ then $P_y = -e^x \sin y$ and $Q_x = -e^x \sin y$ which are identical. So the vector field is conservative and an $f$exists such that

$f_x = P = e^x \cos y$ and $f_y = Q = - e^x\sin y$ which gives $f = e^x \cos y + c$

5. for the field is conservative $\frac {di}{dy} - \frac{dj}{dx} = 0$?

field conservative = field gradient ?

If $f_x = P = e^x \cos y$ and $f_y = Q = - e^x\sin y$ why $f = e^x \cos y + c$ ???

6. Originally Posted by Apprentice123
for the field is conservative $\frac {di}{dy} - \frac{dj}{dx} = 0$?

field conservative = field gradient ?

If $f_x = P = e^x \cos y$ and $f_y = Q = - e^x\sin y$ why $f = e^x \cos y + c$ ???

First since the vector field is conservative you know an $f$ exists, i.e. $f$ exists such that $f_x = P = e^x \cos y \;{\bf and} \;f_y = Q = - e^x\sin y$

If $f_x = e^x \cos y$ then integrating wrt to $x$ gives $f = e^x \cos y + g(y)$. Now subs. into $f_y = - e^x\sin y$ which shows that $- e^x\sin y + g'(y) = - e^x\sin y$ which gives $g'(y) = 0$ which gives $g(y)=c$, thus our $f.$

7. Thank you I understand.

And the statements are correct?

for that the field is conservative i need have => $\frac {di}{dy} - \frac{dj}{dx} = 0$

and
field conservative = field gradient ?