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I'm working through my rheology notes and there is a fairly basic part that I dont understand at all.

If you have two neighbouring points in a liquid P(x,y,z) and $$Q(x + \Delta{x}, y + \Delta{y}, z + \Delta{z})$$ now if the velocity components of P are given as (u, v, w) then $$u_Q = u_P + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{\delta{u}}{\delta{y}}\Delta{y} + \frac{\delta{u}}{\delta{z}}\Delta{z}$$ which i understand its a quite simple derivation and we have similar equations of v & w

But then he just goes on to say

"Then $$u_Q = u_P + \frac{1}{2}(\frac{\delta{u}}{\delta{z}} - \frac{\delta{w}}{\delta{x}})\Delta{z} - \frac{1}{2}(\frac{\delta{v}}{\delta{x}}-\frac{\delta{u}}{\delta{y}})\Delta{y} + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{1}{2}(\frac{ \delta {u}}{\delta {y}}+\frac{\delta {v}}{ \delta {x} })\Delta{y} + \frac{1}{2}(\frac{\delta{u}}{\delta {z}}+\frac{ \delta {w}}{\delta{x}})\Delta{z}$$"

which i dont understand what this is let alone how it was derived I also couldn't find it anywhere on the net

Cheers for any insight you may have

If you have two neighbouring points in a liquid P(x,y,z) and $$Q(x + \Delta{x}, y + \Delta{y}, z + \Delta{z})$$ now if the velocity components of P are given as (u, v, w) then $$u_Q = u_P + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{\delta{u}}{\delta{y}}\Delta{y} + \frac{\delta{u}}{\delta{z}}\Delta{z}$$ which i understand its a quite simple derivation and we have similar equations of v & w

But then he just goes on to say

"Then $$u_Q = u_P + \frac{1}{2}(\frac{\delta{u}}{\delta{z}} - \frac{\delta{w}}{\delta{x}})\Delta{z} - \frac{1}{2}(\frac{\delta{v}}{\delta{x}}-\frac{\delta{u}}{\delta{y}})\Delta{y} + \frac{\delta{u}}{\delta{x}}\Delta{x} + \frac{1}{2}(\frac{ \delta {u}}{\delta {y}}+\frac{\delta {v}}{ \delta {x} })\Delta{y} + \frac{1}{2}(\frac{\delta{u}}{\delta {z}}+\frac{ \delta {w}}{\delta{x}})\Delta{z}$$"

which i dont understand what this is let alone how it was derived I also couldn't find it anywhere on the net

Cheers for any insight you may have

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