div F and curl F

Quick question:

A quest states:

In exercises 1-11, calculate div F and curl F for the given vector fields.

11. F = θ̂ (with a "hat" on top) = -sinθi + cosθj.

If I rewrite F in terms of cartesian coordinates I get:

-(y/(√(x² + y²)) + (x/√(x² + y²))

Then by differentiation followed up by addition as the devergence theorem says I get anything but 0, which is the correct result for this task. How come? Have I not converted the polar coordinates into cartesian coordinates correctly? If I can solve this problem, finding the curl of F shouldn't be that hard.

Quote:

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Originally Posted by kaemper

Quick question:

A quest states:

In exercises 1-11, calculate div F and curl F for the given vector fields.

11. F = θ̂ (with a "hat" on top) = -sinθi + cosθj.

If I rewrite F in terms of cartesian coordinates I get:

-(y/(√(x² + y²)) + (x/√(x² + y²))

Then by differentiation followed up by addition as the devergence theorem says I get anything but 0, which is the correct result for this task. How come? Have I not converted the polar coordinates into cartesian coordinates correctly? If I can solve this problem, finding the curl of F shouldn't be that hard.

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$F=\{ \dfrac{-y}{\sqrt{x^2+y^2}}, \dfrac{x}{\sqrt{x^2+y^2}},0 \}$

$\dfrac{\partial}{\partial x}F_x=\dfrac{x y}{\left(x^2+y^2\right)^{3/2}}$

$\dfrac{\partial}{\partial y}F_y=\dfrac{-x y}{\left(x^2+y^2\right)^{3/2}}$

$\dfrac{\partial}{\partial z}F_z=0$

$\nabla\cdot F=\dfrac{x y}{\left(x^2+y^2\right)^{3/2}}+\dfrac{-x y}{\left(x^2+y^2\right)^{3/2}}+0=0$