- #1

Saketh

- 261

- 2

**Problem**

The velocity of a two-dimensional flow of liquid is given by

[tex]

\textbf{V} = \textbf{i}u(x, y) - \textbf{j}v(x, y).

[/tex]

If the liquid is incompressible and the flow is irrotational show that

[tex]

\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}

[/tex]

and

[tex]

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

[/tex]

\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

[/tex]

**My Work**

I evaluated [itex]\nabla \times \textbf{V} = 0[/itex] through a determinant, and ended up with this expression:

[tex]\textbf{i}\frac{\partial v}{\partial z} + \textbf{i}\frac{\partial u}{\partial z} - \textbf{k}\left ( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial z} \right ) = 0

[/tex]

[/tex]

Through this, I was able to verify:

[tex]

\frac{\partial u}{\partial y} = -\frac{\partial{v}}{\partial x}

[/tex]

\frac{\partial u}{\partial y} = -\frac{\partial{v}}{\partial x}

[/tex]

I could not verify the other expression. How can I verify the other expression - I've tried everything I can think of. It seems simple, but I am missing something.

Thanks in advance.

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