線形代数の記事

$The second-order Cauchy stress tensor T {\displaystyle \mathbf {T} } describes the stress experienced by a material at a given point. For any unit vector v {\displaystyle \mathbf {v} } , the product T ⋅ v {\displaystyle \mathbf {T} \cdot \mathbf {v} } is a vector, denoted T ( v ) {\displaystyle \mathbf {T} (\mathbf {v} )} , that quantifies the force per area along the plane perpendicular to v {\displaystyle \mathbf {v} } . This image shows, for cube faces perpendicular to e 1 , e 2 , e 3 {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}} , the corresponding stress vectors T ( e 1 ) , T ( e 2 ) , T ( e 3 ) {\displaystyle \mathbf {T} (\mathbf {e} _{1}),\mathbf {T} (\mathbf {e} _{2}),\mathbf {T} (\mathbf {e} _{3})} along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor.$

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A linear system in three variables determines a collection of planes. The intersection point is the solution.

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$v {\displaystyle \mathbf {v} } is the linear combination of vectors u 1 {\displaystyle \mathbf {u} _{1}} and u 2 {\displaystyle \mathbf {u} _{2}} such that v = 2 ⋅ u 1 + 1.5 ⋅ u 2 {\displaystyle \mathbf {v} =2\cdot \mathbf {u} _{1}+1.5\cdot \mathbf {u} _{2}}$

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