Respuesta :
The given field is
[tex]f(x,y) = (y \ e^{x} + sin(y) \hat{i} + (e^{x}+x \, cosy) \hat{j}[/tex]
Note that if the field is of the form
[tex]\vec{f} = P\hat{i} + Q\hat{j}[/tex]
then the condition for the field to be conservative is that
[tex] \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} [/tex]
For the given field,
[tex] \frac{\partial P}{\partial y} =e^{x} + cos y \\ \frac{\partial Q}{\partial x} = e^{x}+cosy [/tex]
Because the condition is satisfied, the given field is conservative.
To find the function that satisfies
[tex]f= \bigtriangledown f[/tex]
[tex]f(x,y) = \int P(x,y)dx = ye^{x} + x\, siny + g(y)[/tex]
Because [tex] \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} [/tex], therefore
[tex] \frac{\partial f}{\partial y} =e^{x} + x \, siny + g'(y)[/tex]
Because [tex] \frac{\partial f}{\partial y} =Q = e^{x} + x \, cosy [/tex]
therefore
g'(y) = 0
g = c (constant).
Answer:
(a) The given function is conservative.
(b) The function is
[tex]f(x,y) = ye^{x} + x \, siny + c[/tex]
where c = constant.
[tex]f(x,y) = (y \ e^{x} + sin(y) \hat{i} + (e^{x}+x \, cosy) \hat{j}[/tex]
Note that if the field is of the form
[tex]\vec{f} = P\hat{i} + Q\hat{j}[/tex]
then the condition for the field to be conservative is that
[tex] \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} [/tex]
For the given field,
[tex] \frac{\partial P}{\partial y} =e^{x} + cos y \\ \frac{\partial Q}{\partial x} = e^{x}+cosy [/tex]
Because the condition is satisfied, the given field is conservative.
To find the function that satisfies
[tex]f= \bigtriangledown f[/tex]
[tex]f(x,y) = \int P(x,y)dx = ye^{x} + x\, siny + g(y)[/tex]
Because [tex] \frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x} [/tex], therefore
[tex] \frac{\partial f}{\partial y} =e^{x} + x \, siny + g'(y)[/tex]
Because [tex] \frac{\partial f}{\partial y} =Q = e^{x} + x \, cosy [/tex]
therefore
g'(y) = 0
g = c (constant).
Answer:
(a) The given function is conservative.
(b) The function is
[tex]f(x,y) = ye^{x} + x \, siny + c[/tex]
where c = constant.
