Question

Write out the sums${\mathit{P}}_{\mathbf{i}\mathbf{j}}{\mathit{e}}_{\mathbf{j}}$ for each value of and compare the discussion of $\mathbf{(}\mathbf{1}\mathbf{.1}\mathbf{)}$.Hint: For example, if$\mathit{i}\mathbf{=}\mathbf{2}$ [or y in$\mathbf{(}\mathbf{1}\mathbf{.1}\mathbf{)}$ ], then the pressure across the face perpendicular to the${\mathit{x}}_{\mathbf{2}}$axis is $P_{21}{e}_1+P_{22}{e}_{22}+P_{23}{e}_3$, or, in the notation of (1.1), ${\mathit{P}}_{\mathbf{y}\mathbf{x}}\mathit{i}\mathbf{+}{\mathit{P}}_{\mathbf{y}\mathbf{y}}\mathit{j}\mathbf{+}{\mathit{P}}_{\mathbf{y}\mathbf{z}}\mathit{k}$.

Short answer

The total force per unit area acting on the surface perpendicular to the $i^{th}$- axis .

$P_{ij}{e}_j=P_{ix}i+P_{iy}j+P_{iz}k$

Step-by-step solution

Given Information

Force per unit area $P_{ij}$ is perpendicular to the $i^{th}$direction and acting in the $j^{th}$direction.

Definition of a force per unit area.

The force exertedon a unit area of the surface is known as force per unit area.

Find the sum.

Given, Force per unit area$P_{ij}$is perpendicular to the $i^{th}$direction and acting in the $j^{th}$direction.

Therefore, the total force per unit area acting on the surface perpendicular to the $i^{th}$- axis is mentioned below.

$P_{ij}{e}_j=P_{ix}i+P_{iy}j+P_{iz}k$