Explanation:

The symmetry of the stress tensor at a point in a body when at equilibrium is obtained from the moment equilibrium equations. The stress tensor is a mathematical representation of the distribution of forces within a material. It describes the internal forces acting on an infinitesimal element within the material.

Moment Equilibrium Equations:
The moment equilibrium equations are derived from the principles of statics and state that the sum of the moments acting on a body must be zero for equilibrium. These equations are derived by considering the equilibrium of forces and moments acting on an infinitesimal element within the material.

Symmetry of Stress Tensor:
The moment equilibrium equations are crucial in determining the symmetry of the stress tensor. Symmetry refers to the balance or equality of the stress components in different directions. In the absence of any external moments or couples, the stress tensor is symmetric.

Explanation of Symmetry:
The symmetry of the stress tensor can be understood by considering the equilibrium of moments. When an infinitesimal element within a body is in equilibrium, the sum of the moments acting on it must be zero in all directions. This implies that the shear stresses acting on opposite faces of the element must be equal and opposite.

Example:
For example, consider a cube-shaped element within a material. If the material is in equilibrium, the shear stresses on opposite faces of the cube must be equal and opposite. Similarly, the normal stresses acting on opposite faces of the cube must also be equal.

Importance of Symmetry:
The symmetry of the stress tensor is important as it allows for simplification of calculations and analysis. By assuming symmetry, engineers and scientists can make certain assumptions and simplify equations, making it easier to analyze the behavior of materials under different loading conditions.

Conclusion:
In conclusion, the symmetry of the stress tensor at a point in a body when at equilibrium is obtained from the moment equilibrium equations. These equations ensure that the shear and normal stresses acting on opposite faces of an infinitesimal element within the material are equal and opposite, leading to a symmetric stress tensor.