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Unformatted text preview: NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Revision Notes (Multi-variable Calculus) 7 Summarization (Tutorial 6 - 10) Tutorial 6-9 is concerning about multi-variable functions (Scalar Fields and Vector Fields) 7.1 Some Derivatives 1. Partial Derivatives – The easiest multi-variable derivatives. For example ∂ ∂x f ( x, y, z ), we regard y and z as constants, and then take partial derivative with respect to x is the same as take derivative df ( x ) dx . – Partial derivatives can not be regarded as quotients. – What is the chain rule for multi-variable functions? Suppose the function is given by f ( x, y, z ), then the chain rule says: df dt = ∂f ∂x · dx dt + ∂f ∂y · dy dt + ∂f ∂z · dz dt 2. Gradient Field – Definition: ∇ f = ⎛ ⎜ ⎜ ⎜ ⎝ ∂ ∂x f ∂ ∂y f ⎞ ⎟ ⎟ ⎟ ⎠ – Physical Meaning (Tutorial 6 Question 3) Gradient is the direction that gives the largest directional derivative. Remark: This is a conclusion of the tutorial 6 question 3, and we can use it as a short-cut to find the direction of largest directional derivative. Check Thomas’ Calculus, Section 14.5, Questions 17-22, and their solutions. 3. Directional Derivative – Please remember that, we have two kinds of notations for directional derivative: for the directional derivative of the function f ( x, y ) at the point ( x , y ) along the direction u can be written as: df dx u , ( x ,y ) vs D u f ( x , y ) – The direction u must be UNIT , i.e., the magnitude of u is 1! 234 – Directional derivative is a linear combination of all the partial derivatives, and the co- eﬃcients are determined by the direction u . With the notation gradient, we can write the directional derivative as: (This is the method to get the directional derivative.) D u f ( x , y ) = ∇ f ( x , y ) · u . 4. Operators: ∇ , curl and div if we regard the notation ∇ as an operator on functions, we many have three kinds results: Gradient : f = ∂f ∂x i + ∂f ∂y j + ∂f ∂z k Divergence : · F = ∂P ∂x + ∂Q ∂y + ∂R ∂z = div F ; Curl : × F = ∂R ∂y − ∂Q ∂z i + ∂P ∂z − ∂R ∂x j + ∂Q ∂x − ∂P ∂y k = curl F . The first one is for scalar functions, the second and third are for vector fields (the second one is dot product, the third one is cross product). 5. Applications The significant application of partial derivatives are used to find the the critical points, local maxima, local minima and saddle points. – Critical points: DEFINITION Critical Point An interior point of the domain of a function ƒ ( x , y ) where both and are zero or where one or both of and do not exist is a critical point of ƒ . ƒ y ƒ x ƒ y ƒ x Pay special attention to the following words: An interior point ······ where both f x and f y are zero or where one or both of f x and f y do not exist ······ ....
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- Spring '09