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# If X = Uv, Y = U + V U − V . Find ∂ ( U , V ) ∂ ( X , Y ) . - Applied Mathematics 1

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ConceptReview of Complex Numbers‐Algebra of Complex Number

#### Question

If x = uv, y =(u+v)/(u-v).find (del(u,v))/(del(x,y)).

#### Solution

(del(u,v))/(del(x,y))= $\begin{vmatrix}u_x & u_y \\ v_x & v_y\end{vmatrix}$ =$\begin{vmatrix} \frac{\delta x}{\delta u} & \frac{\delta x}{\delta v} \\ \frac{\delta y}{\delta u} & \frac{\delta y}{\delta v}\end{vmatrix}$

(del x)/(del u)=del(uv)=v...................(2)

(delx)/(delv)=del(uv)=u.....................(3)

(dely)/(delu)=del((u+v)/(u-v))=((u-v)-(u+v))/((u-v)^2)=(-2v)/(u-v)^2...............(4)

(dely)/(delv)=del((u+v)/(u-v))=((u-v)+(u+v))/(u-v)^2=(2u)/((u-v)^2 ....................(5)

From equation (2),(3),(4),(5) we get,

$\begin{vmatrix} \frac{\delta x}{\delta u} & \frac{\delta x}{\delta v} \\ \frac{\delta y}{\delta u} & \frac{\delta y}{\delta v}\end{vmatrix}$ = $\begin{vmatrix} v & u \\ \frac {-2v} {u-v}^2 & \frac {2u} {uv} \end{vmatrix}$ = (2uv)/(u-v)^2+(2uv)/(u-v)^2=(4uv)/(u-v)^2

From (1) we get,

JJ’=1

Jxx(4uv)/(u-v)^2=1 .........................(let J'=(4uv)/(u-v)^2)

Hence J = (u-v)^2/(4uv)

therefore e^(2varphi)=cot (alpha/2)

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#### APPEARS IN

Solution If X = Uv, Y = U + V U − V . Find ∂ ( U , V ) ∂ ( X , Y ) . Concept: Review of Complex Numbers‐Algebra of Complex Number.
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