Solve any maths problem

Step by step

1. T = x^{2} y^{2},\quad \mathbf{u} = \left\langle - y,\ x,\ 0 \right\rangle,\quad \frac{D}{Dt} = \frac{\partial}{\partial t} + - y\,\frac{\partial }{\partial x} + x\,\frac{\partial }{\partial y} + 0\,\frac{\partial }{\partial z}

   The material derivative follows a fluid particle: the change at a fixed point, plus the change from being carried along by the flow.

2. \frac{\partial T}{\partial t} = 0

   The local rate of change: differentiate with respect to t at a fixed position.

3. \frac{\partial T}{\partial x} = 2 x y^{2}

   Differentiate with respect to x, holding t and the other coordinates constant.

4. \frac{\partial T}{\partial y} = 2 x^{2} y

   Differentiate with respect to y, holding t and the other coordinates constant.

5. \frac{\partial T}{\partial z} = 0

   Differentiate with respect to z, holding t and the other coordinates constant.

6. \mathbf{u}\cdot\nabla T = \left(- y\right)\left(2 x y^{2}\right) + \left(x\right)\left(2 x^{2} y\right) + \left(0\right)\left(0\right) = 2 x^{3} y - 2 x y^{3}

   The convective part: each velocity component times the matching slope.

7. \frac{DT}{Dt} = 0 + \left(2 x^{3} y - 2 x y^{3}\right) = 2 x y \left(x^{2} - y^{2}\right)

   Add the local and convective parts.