MHB Where Are the Points Satisfying All Conditions in This Vector Field?

[Drain Brain]

A vector field is specified as F= $2(x+y)\sin(\pi z)a_x-(x^2+y)a_y+\left(\frac{10}{(x^2+y^2)}\right)a_z$. Specify the locus of all points at which: a. $F_x=0$; b. $F_y=0$; c. |F|=1


Please kindly help me to get started with this problem! thanks!

 

[Prove It]

A vector field is specified as F= $2(x+y)\sin(\pi z)a_x-(x^2+y)a_y+\left(\frac{10}{(x^2+y^2)}\right)a_z$. Specify the locus of all points at which: a. $F_x=0$; b. $F_y=0$; c. |F|=1


Please kindly help me to get started with this problem! thanks!

Have you at least tried to work out the partial derivatives?

 

[Drain Brain]

Have you at least tried to work out the partial derivatives?

partial derivative of what? please bear with me.

 

[Prove It]

partial derivative of what? please bear with me.

The question clearly says you need $\displaystyle \begin{align*} F_x \end{align*}$ and $\displaystyle \begin{align*} F_y \end{align*}$...

 

[chisigma]

A vector field is specified as F= $2(x+y)\sin(\pi z)a_x-(x^2+y)a_y+\left(\frac{10}{(x^2+y^2)}\right)a_z$. Specify the locus of all points at which: a. $F_x=0$; b. $F_y=0$; c. |F|=1


Please kindly help me to get started with this problem! thanks!

Because...

$\displaystyle F_{x} = 2\ (x+y)\ \sin (\pi\ z)\ a_{x} $

$\displaystyle F_{y} = - (x^{2} + y)\ a_{y}$

$\displaystyle F_{z} = \frac{10}{x^{2}+ y^{2}}\ a_{z}\ (1)$

... where $a_{x}$, $a_{y}$ and $a_{z}$ are the versors along the cartesian axes, the response to pints a. and b. is immediate...

... the response of the point c. starts from the relation...

$\displaystyle 4\ (x + y)^{2}\ \sin^{2} (\pi\ z) + (x^{2} + y)^{2} + \frac{100}{(x^{2}+y^{2})^{2}} = 1\ (2)$

Kind regards

$\chi$ $\sigma$

 

[Drain Brain]

Because...

$\displaystyle F_{x} = 2\ (x+y)\ \sin (\pi\ z)\ a_{x} $

$\displaystyle F_{y} = - (x^{2} + y)\ a_{y}$

$\displaystyle F_{z} = \frac{10}{x^{2}+ y^{2}}\ a_{z}\ (1)$

... where $a_{x}$, $a_{y}$ and $a_{z}$ are the versors along the cartesian axes, the response to pints a. and b. is immediate...

... the response of the point c. starts from the relation...

$\displaystyle 4\ (x + y)^{2}\ \sin^{2} (\pi\ z) + (x^{2} + y)^{2} + \frac{100}{(x^{2}+y^{2})^{2}} = 1\ (2)$

Kind regards

$\chi$ $\sigma$

Do I have to take the partial derivative of $F_{x}$ and $F_{y}$ to specify the locus of points?

 

[chisigma]

Do I have to take the partial derivative of $F_{x}$ and $F_{y}$ to specify the locus of points?

I don't think so!... in my previous post explicit expressions for $F_{x}$ and $F_{y}$ have been given, so that the loci of points are obtained imposing $F_{x}=0$ and $F_{y}=0$... the last case is very easy because...

$\displaystyle F_{y} = 0 \implies x^{2} + y = 0 \implies y = - x^{2}\ (1)$

... and the locus is a parabola...

Kind regards

$\chi$ $\sigma$

 

[Drain Brain]

I don't think so!... in my previous post explicit expressions for $F_{x}$ and $F_{y}$ have been given, so that the loci of points are obtained imposing $F_{x}=0$ and $F_{y}=0$... the last case is very easy because...

$\displaystyle F_{y} = 0 \implies x^{2} + y = 0 \implies y = - x^{2}\ (1)$

... and the locus is a parabola...

Kind regards

$\chi$ $\sigma$

So for
$\displaystyle F_{x} = 0 \implies 2(x+y)\sin(\pi z)= 0 \implies y = - x ( plane)$

But how do I do that for $|F| =1$?

 

[chisigma]

So for
$\displaystyle F_{x} = 0 \implies 2(x+y)\sin(\pi z)= 0 \implies y = - x ( plane)$

But how do I do that for $|F| =1$?

Is $F_{x} = 0$ not only fon the plane $y = - x$, but also for the planes $\sin (\pi z) = 0 \implies z= k \in \mathbb {Z}$...

Regarding $|F| = 1$ the locus is defined by the relation...

$\displaystyle F_{x}^{2} + F_{y}^{2} + F_{z}^{2} = 1 \implies 4\ (x+y)^{2}\ \sin^{2} (\pi z) + (x^{2} + y)^{2} + \frac{100}{(x^{2} + y^{2})^{2}} = 1$

Kind regards

$\chi$ $\sigma$

 

[Drain Brain]

Is $F_{x} = 0$ not only fon the plane $y = - x$, but also for the planes $\sin (\pi z) = 0 \implies z= k \in \mathbb {Z}$...

Regarding $|F| = 1$ the locus is defined by the relation...

$\displaystyle F_{x}^{2} + F_{y}^{2} + F_{z}^{2} = 1 \implies 4\ (x+y)^{2}\ \sin^{2} (\pi z) + (x^{2} + y)^{2} + \frac{100}{(x^{2} + y^{2})^{2}} = 1$

Kind regards

$\chi$ $\sigma$

The answer in my book says $x^2+y^2=10$ i have no idea how it arrived ti that answer. Please tell me how to go about it.

 

[chisigma]

The answer in my book says $x^2+y^2=10$ i have no idea how it arrived ti that answer. Please tell me how to go about it.

Of course under the conditions $F_{x}=0$ and $F_{y}=0$ is $|F|=1$ if and only if $x^{2} + y^{2} = 10$...

Kind regards

$\chi$ $\sigma$