Implicit differentiation and the inverse Jacobian
| English | Français |
|---|---|
| local inverse/ˈləʊkl ɪnˈvɜːs/ | local inverse |
| Jacobian matrix/dʒæˈkəʊbɪən ˈmeɪtrɪks/ | Jacobian matrix |
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?
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?
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 ____.
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.
Which answer fits this case?
Differentiate a coupled implicit system as a linear system
A zero Jacobian determinant proves that the underlying map has no inverse of any kind.
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.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.