Is this vector calculus notation correct?

[hotvette]

My vector calculus is a bit rusty. Can anyone tell me if the following uses proper symbolism?

$$F &= \left[\begin{matrix}f_1(x_1,x_2) \\ f_2(x_1,x_2) \\ f_3(x_1,x_2) \end{matrix}\right] \qquad x = \left[\begin{matrix} x_1 \\ x_2 \end{matrix}\right] \qquad \frac{DF}{dx}&= \left[\begin{matrix} \rule{0pt}{3ex}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\ \rule{0pt}{3ex}\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \\ \rule{0pt}{3ex}\frac{\partial f_3}{\partial x_1} & \frac{\partial f_3}{\partial x_2}\end{matrix}\right]$$

 

[yyat]

The first two look correct, but I don't think $$\frac{DF}{dx}$$ is a common notation. I would use just $$DF$$ or maybe $$\frac{\partial f}{\partial x}$$.

Edit: http://en.wikipedia.org/wiki/Jacobian_matrix" uses the notation $$J_F$$.

 

[Jon89bon]

Im very turned around in vector calc! I have the equation 4y² -4z²=5y/x-4x² and I need to convert it from rectangular to cylindrical coordinates. Could someone explain this better than my professor?

 

[Marin]

Hi, Jon89bon!

I had a bit of vector calculus in my first semester at the university and I'm not sure if what I paste here is truly correct, so it would need an overview from a supervisor :)

Here's what we do:

1. we transform the equation: 4y^2 -4z^2=5y/x-4x^2 into

(4y^2 -4z^2+4x^2)x-5y=0=:f(x,y,z) and set a func. f equal to it.

2. As f is a SCALAR field, we could use the transformations:

(x,y,z)=(rcos(a),rsin(a),z), where r^2=x^2+y^2 and a is the angle of rotation around the z-axisremark: if you're good at algebra, you could skip the first step :)IMPORTANT: I think, this transformation does not apply to vector fields (when f is a vector), but I need an approval for that statement from s.o. else :)all the best, marin

 

[Jon89bon]

Thanks I will work with this and see what I come up with!

Jon89bon