Question

04. Prove that Var 1/-e + + m te, I in cylindrical coordinates. in upberical coordinates. Orrekin

Answer 1

These relations can be easily proven by using general formula for gradient in orthogonal curvilinear coordinates which is given by:

1 afat Jest hz Jur vf= hou 1 01. én + haus 1 af

Where
U1, U2, U:
are orthogonal curvilinear coordinates ,
ei, 62, 63
are the unit vectors in the direction of
U1, U2, U:
,

and $h_1$ is the magnitude of tangent vectors to the curve $u_1$ at a point where $u_2$ and $u_3$ are constant i.e
hi = Out
, similarly
др h2 = ди
and
h3 = диз
, here $\vec{p}$ is the position vector of the point with curvilinear

coordinates $(u_1,u_2,u_3)$.

Proof 1:

Let $\vec{p}$ be the position vector of point P, therefore we have $\vec{p}=x\hat{i}+y\hat{j}+z\hat{k}$

Now as in cylindrical coordinates $x=rcos\theta,y=rsin\theta,z=z$ therefore we have

p=rcosti + rsinaj + zk

Now to apply the above formula we need to find $h_1,h_2,h_3$ . Which are as follows:

$h_1=\left |\frac{\partial \vec{p}}{\partial r}\right |=\sqrt{cos^2\theta+sin^2\theta}=1$

$h_2=\left |\frac{\partial \vec{p}}{\partial \theta}\right |=\sqrt{r^2(sin^2\theta+cos^2\theta)}=r$

$h_3=\left |\frac{\partial \vec{p}}{\partial z}\right |=1$

and in cylindrical coordinates the unit vectors will be

ér, ég, és

Substituting these values in the general formula for gradient in the curvilinear coordinates we get the required result:

Vf - Ofe 10f Ofer + of 3460 + oz.

Proof 2:

Let $\vec{p}$ be the position vector of point P, therefore we have $\vec{p}=x\hat{i}+y\hat{j}+z\hat{k}$

Now as in spherical polar coordinates therefore we have

x=rsin coso, y = rsino sino, z = rcos

$\vec{p}=rsin\theta cos\phi \hat{i}+rsin\theta sin\phi\hat{j}+rcos\theta\hat{k}$

Now to apply the above formula we need to find $h_1,h_2,h_3$ . Which are as follows:

21 = sin cosoi + sint sinoj + sincos-o + sin-o sin-o + cose

..h1 = V sin20(cos-o + sin26) + cos26 = 1

rcostcosoi +rcost sinoj-rsinek

:. h2 = r2(cos20(cos + sinºn) + cos20) = 1

h3 = -r sino sinoi +rsin cosoi

:. h3 == r2 sin20 (sin6 + cos26) = rsino

in spherical polar coordinates the unit vectors will be

ê, ê, eo

Substituting these values in the general formula for gradient in the curvilinear coordinates we get the required result:

er of lof V = ore tra 1 of + rsino a zº

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