The discussion revolves around identifying the locus of points in a specified vector field, defined as F= $2(x+y)\sin(\pi z)a_x-(x^2+y)a_y+\left(\frac{10}{(x^2+y^2)}\right)a_z$. Participants explore conditions where the components of the vector field are zero and where the magnitude of the vector field equals one. The scope includes mathematical reasoning and technical exploration of vector calculus.
Participants express differing views on the necessity of taking partial derivatives to specify the locus of points. While some agree on the derived equations for $F_x=0$ and $F_y=0$, the discussion remains unresolved regarding the derivation of the condition $|F|=1$ and its relation to the book's answer.
There are unresolved mathematical steps regarding the derivation of the condition $|F|=1$ and how it relates to the given answer in the book. The discussion also reflects varying interpretations of the conditions under which the vector field components equal zero.
Drain Brain said: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 said:Have you at least tried to work out the partial derivatives?
Drain Brain said:partial derivative of what? please bear with me.
Drain Brain said: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!
chisigma said: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 said:Do I have to take the partial derivative of $F_{x}$ and $F_{y}$ to specify the locus of points?
chisigma said: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 said: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 said: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 said: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.