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Unformatted text preview: Multivariable calculus
Partial derivatives are needed when we Partial have more variables – they measure changes of f(x,y,z,…) with respect to x,y,z,…, one at a time while the others are fixed. Ordinary derivative only deals with Ordinary change in f(x) w.r.t. a single variable x Matrix connection: Jacobian and Matrix Hessian
63 63 Taylor’s Theorem
Single-variable: linear approximation Single using ordinary derivative, mean-value theorem, the remainder, and examples (find sqrt(10)) Multivariable extension seems not trivial Multivariable One function f(x,y) and two variables x,y One is the simplest multivariable extension Linear approximation: f(x+h,y+k) = Linear f(x,y)+( h fx(x,y)+k fy(x,y) )
64 Hessian matrix (symmetric matrix of second derivatives)
What is the error using the linear What approximation? Hessian matrix is the answer. Why? Hessian Better notation for Taylor’s Theorem Better using vectors and matrices
• Using gradient vector (or “Nabla”) in the linear approximation • Using Hessian in the error term (quadratic form) 65 65 Where is the Jacobian (matrix of first derivatives)?
The Jacobian matrix is the (gradient The vector)T for the single function case A single function and n variables single
• Jacobian matrix has dimensions 1xn • Hessian matrix has dimensions nxn m functions and n variables functions
• Jacobian dimensions change to mxn • Hessian dimensions are still nxn
66 Vector calculus
Understanding the “Nabla” and Understanding “Laplacian” Partial derivatives are scalars but Nabla Partial (or gradient) is a vector Laplacian is the gradient of gradient but Laplacian it is a scalar Putting specified direction to derivatives: Putting Directional derivatives
67 67 Directional derivatives
Finding changes in specified direction of Finding a unit vector v Directional derivatives should be the Directional same under different coordinate systems Example: gradient in Cartesian Example: coordinates and gradient in cylindrical coordinates. Will they give the same directional derivative for the same v? 68 Relating Jacobians using chain rule
What is the chain rule in the simplest What form? Relating gradients using Jacobian Relating (where are the chain rules?) Generalized chain rule to relate Generalized Jacobian matrices 69 69 Different roles of Jacobian
Linear model for small changes Linear (Taylor’s Theorem) Chain rule via Jacobian for change of Chain coordinates Area changes from dxdy to drdθ using Area Jacobian (determinant) 70 Critical points (or stationary points)
Multivariable function – n variables Multivariable Existence of critical points: zero first Existence derivatives Classification of critical points: Classification properties of second derivatives The role of Hessian in the classfication The 71 71 Hessian in optimization
One variable case: example One Two variable case: example Two n-variable case Hessian: Positive-definite, negativeHessian: definite, and eigenvalues Local minima, local maxima, and saddle Local 72 ...
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- Spring '10