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 Calculus Calculus Math Forum

 April 8th, 2019, 10:42 AM #1 Newbie   Joined: Mar 2019 From: - Posts: 3 Thanks: 0 Implicit Differentiation Given is the function F(x,y,z) = x^2+y^3-z. Determine the Jacobian matrix Dz in P=(1,1,2) using implicit differentiation. My idea is to calculate ∂z/∂x in P(1,1,2) and ∂z/∂y in P(1,1,2) and then just write it in matrix form. So, F(x,y,z)=x^2+y^3-z=0 ∂z/∂x=-((∂F/∂x)/(∂F/∂z))=-(2x/-1)=2x ∂z/∂x in P(1,1,2)=2 ∂z/∂y=-((∂F/∂y)/(∂F/∂z))=-(3y2/-1)= 3y^2 ∂z/∂y in P(1,1,2)=3 Dz (1,1,2) = (∂z/∂x(1,1,2) ∂z/∂y(1,1,2)), Dz is [1 X 2] matrix Dz (1,1,2) = (2 3) I checked the result using explicit differentiation and I obtained the same. But in the book that I use I saw another approach. Namely, as a hint was given this formula: Dx=f(x°)=-[DyF(x°,y°)]-1DxF(x°,y°) I don’t understand how this formula can be used in order to calculate Dz. Any help is appreciated. Thanks in advance. April 9th, 2019, 12:56 PM #2 Newbie   Joined: Mar 2019 From: - Posts: 3 Thanks: 0 It seems that both approaches are indeed equivalent. Since Dz is requested, Dx F(x°)=-[Dy F(x°,y°)]^(-1)Dx F(x°,y°) becomes DxyF(x°,y°)=-[Dz F(x°,y°,z°)]^(-1) Dxy F(x°,y°,z°) It follows that: -[Dy F(x°,y°)]^(-1)=-[-1]^(-1) Dxy F(x°,y°,z°)=(2x 3x^2) Dxy F(1,1,2)=(2 3) So, Dz = -[-1]^(-1) (2 3) = (2 3) Tags differentiation, implicit, implicit differentiation, implicit function theorem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Omnipotent Calculus 3 October 2nd, 2014 07:12 PM Sonprelis Calculus 2 May 1st, 2014 04:11 AM mathkid Calculus 2 September 25th, 2012 11:27 AM mathkid Calculus 1 September 25th, 2012 08:26 AM ohspyro89 Calculus 4 October 6th, 2008 05:48 AM

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