ベクトル空間の記事

$A homogeneous function is not necessarily continuous, as shown by this example. This is the function f {\displaystyle f} defined by f ( x , y ) = x {\displaystyle f(x,y)=x} if x y > 0 {\displaystyle xy>0} and f ( x , y ) = 0 {\displaystyle f(x,y)=0} if x y ≤ 0. {\displaystyle xy\leq 0.} This function is homogeneous of degree 1, that is, f ( s x , s y ) = s f ( x , y ) {\displaystyle f(sx,sy)=sf(x,y)} for any real numbers s , x , y . {\displaystyle s,x,y.} It is discontinuous at y = 0 , x ≠ 0. {\displaystyle y=0,x\neq 0.}$

同次関数（斉次関数）の定義と性質数学的基礎からオイラーの定理まで

同次関数斉次関数正の同次性同次次数

2026年8月11日続きを読む →

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

線形結合の基礎から応用までベクトル空間における合成の仕組み

線形結合ベクトル空間線形代数線形独立

2026年8月9日続きを読む →

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

テンソルとは何か数学的定義から物理学への応用までを解説

テンソル多重線形写像ベクトル空間階数

2026年8月7日続きを読む →