은미 미분 및 역 자코비안
도입됨| English | 의미 |
|---|---|
| local inverse/ˈləʊkl ɪnˈvɜːs/ | 국소 역함수 |
| Jacobian matrix/dʒæˈkəʊbɪən ˈmeɪtrɪks/ | 야코비안 행렬 |
A decision before an answer
- Two output coordinates can each depend on both inputs. Recovering an input sensitivity then requires inverting the whole derivative matrix, rather than reciprocating one entry.
- Your goal: Differentiate a coupled implicit system as a linear system.
Read the relationship
- For a differentiable map F(u,v)=(x,y)=(f(u,v),g(u,v)), its Jacobian is J=[[f_u,f_v],[g_u,g_v]]. Small changes satisfy [dx,dy]ᵀ=J[du,dv]ᵀ to first order. To find u_x while holding y fixed, differentiate both defining equations with respect to x: f_u u_x+f_v v_x=1 and g_u u_x+g_v v_x=0. These are a coupled linear system, not two independent scalar inverse rules.
- Use a nonzero Jacobian determinant to justify a local inverse.
For x=u+v,y=u+2v, what is u_x when y is held fixed?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The inverse matrix of [[1,1],[1,2]] is [[2,−1],[−1,1]]. Its top-left entry is u_x=2.
Use the defining rule
- If f and g are continuously differentiable near the point and det J=f_u g_v−f_v g_u is nonzero there, the inverse-function theorem supplies a differentiable local inverse. Its derivative is J⁻¹=(1/det J)[[g_v,−f_v],[−g_u,f_u]]. Thus u_x=g_v/det J, u_y=−f_v/det J, v_x=−g_u/det J and v_y=f_u/det J. Evaluate every derivative at the corresponding point. A local inverse need not extend to a global one.
- Distinguish inverse partial derivatives from scalar reciprocal rules.
J=[[2,1],[1,−1]]. What is v_y in the inverse derivative matrix?
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det J=−3. The bottom-right inverse entry is 2/(−3)=−2/3.
Check the conditions
- For implicit equations H(u,v,x,y)=0 and K(u,v,x,y)=0, differentiate while fixing the requested independent coordinate. Solve [[H_u,H_v],[K_u,K_v]][u_x,v_x]ᵀ=−[H_x,K_x]ᵀ. The determinant in the unknown variables u,v must be nonzero to use the usual implicit-function theorem. Signs on the right come from moving known derivatives to the other side. If the determinant vanishes, this theorem is inconclusive; it does not by itself prove no inverse or no implicit solution exists.
- Distinguish inverse partial derivatives from scalar reciprocal rules.
Take x=u²+v and y=u−v. At (u,v)=(1,0), the output is (1,1), and J=[[2,1],[1,−1]] has determinant −3. The inverse is [[1/3,1/3],[1/3,−2/3]], so u_x=1/3 and v_y=−2/3 there. Multiplying J by this inverse gives the identity. The reciprocal 1/f_u=1/2 is not u_x because changing u also requires changing v to keep y fixed.
The determinant of J=[[2,1],[1,−1]] is ____.
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Compute 2·(−1)−1·1=−3.
Apply the task format
- The scalar shortcut du/dx=1/(dx/du) holds for a one-variable inverse with nonzero derivative, but usually fails for a coupled system because v changes to keep y fixed. For x=u+v,y=u+2v, J=[[1,1],[1,2]] has determinant 1 and inverse [[2,−1],[−1,1]]. Therefore u_x=2 while 1/f_u=1. An inverse Jacobian transforms differential sensitivities; a change-of-variables integral instead uses the absolute determinant for area or volume scaling, not a selected inverse entry.
- Distinguish inverse partial derivatives from scalar reciprocal rules.
Differentiate both equations and state which output is held fixed. Nonzero determinant guarantees a local inverse under the regularity hypotheses; zero determinant is not a proof of impossibility. Use an inverse entry for sensitivities and an absolute determinant for integration.
다음 중 이 사례에 부합하는 답안은 무엇입니까?
연결된 암시적 시스템을 선형 시스템으로 미분함
A zero Jacobian determinant proves that the underlying map has no inverse of any kind.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
The usual differentiable inverse theorem is inconclusive. For example (u,v)↦(u³,v) is bijective but has zero determinant at u=0 and an inverse not differentiable there.
Keep the distinctions
- Jacobian matrix 雅可比矩阵 — The matrix of first partial derivatives of a vector-valued map.
- local inverse 局部逆映射 — An inverse defined on neighbourhoods of a particular input and output point.
- Differentiate a coupled implicit system as a linear system.
- Use a nonzero Jacobian determinant to justify a local inverse.
- Distinguish inverse partial derivatives from scalar reciprocal rules.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
난이도 3/5 · 200 코인 + 오늘 첫 완료 시 소액 보너스
The order follows the stated process; check each stage before the next.