91Ó°ÊÓ

The divergence of a vector field \( \mathbf{u} = (u_x, u_y, u_z) \) is given by the sum of the partial derivatives of each component with respect to its corresponding coordinate. Mathematically, it is expressed as: \[abla \cdot \mathbf{u} = \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} + \frac{\partial u_z}{\partial z}\]

Compute each partial derivative for the components of \( \mathbf{u} \):1. \( \frac{\partial u_x}{\partial x} = \alpha + 2\beta x \)2. \( \frac{\partial u_y}{\partial y} = \alpha + 2\beta y \)3. \( \frac{\partial u_z}{\partial z} = \alpha + 2\beta z \)

Add the computed partial derivatives to find the divergence:\[abla \cdot \mathbf{u} = (\alpha + 2\beta x) + (\alpha + 2\beta y) + (\alpha + 2\beta z) = 3\alpha + 2\beta(x+y+z)\]

The curl of a vector field \( \mathbf{u} = (u_x, u_y, u_z) \) is calculated using the determinant form:\[abla \times \mathbf{u} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \u_x & u_y & u_z\end{vmatrix}\]This expands to components concerning cyclic permutations of \(x, y, z\).

Compute partial derivatives needed for the determinant:\( \frac{\partial u_z}{\partial y} = 2\alpha \), \( \frac{\partial u_y}{\partial z} = 2\alpha \), \( \frac{\partial u_x}{\partial z} = 0 \), \( \frac{\partial u_z}{\partial x} = 2\alpha \), \( \frac{\partial u_x}{\partial y} = 2\alpha \), \( \frac{\partial u_y}{\partial x} = 0 \).

The components of the curl vector \( abla \times \mathbf{u} \) are:1. \( (abla \times \mathbf{u})_x = 0 - 0 = 0 \)2. \( (abla \times \mathbf{u})_y = 0 - 0 = 0 \)3. \( (abla \times \mathbf{u})_z = 0 - 0 = 0 \)Thus, \( abla \times \mathbf{u} = (0, 0, 0) \).

Cauchy's strain tensor \( \varepsilon \) is given by:\[\varepsilon_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)\]where \(i, j\) represent \(x, y, z\) respectively.

Calculate individual components of the strain tensor:- For \( \varepsilon_{xx}: \frac{1}{2}(\frac{\partial u_x}{\partial x} + \frac{\partial u_x}{\partial x}) = \alpha + \beta x \)- For \( \varepsilon_{yy}: \frac{1}{2}(\frac{\partial u_y}{\partial y} + \frac{\partial u_y}{\partial y}) = \alpha + \beta y \)- For \( \varepsilon_{zz}: \frac{1}{2}(\frac{\partial u_z}{\partial z} + \frac{\partial u_z}{\partial z}) = \alpha + \beta z \)- For \( \varepsilon_{xy}, \varepsilon_{yx}: \frac{1}{2}(\frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x}) = \alpha \)- For \( \varepsilon_{yz}, \varepsilon_{zy}: \frac{1}{2}(\frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y}) = \alpha \)- For \( \varepsilon_{zx}, \varepsilon_{xz}: \frac{1}{2}(\frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z}) = \alpha \)