mathCalculus2-07-draft - ARE211, Fall 2007 LECTURE #17:...

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Preliminary draft only: please check for Fnal version ARE211, Fall 2007 LECTURE #17: THU, OCT 25, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS2) Contents 4. Univariate and Multivariate Diferentiation (cont) 1 4.3. Multivariate calculus: Functions From R n to R 1 4.3.1. Partial Derivative 1 4.3.2. The Gradient 2 4.3.3. Crosspartial Derivative 3 4.3.4. The diferential For Functions From R n to R 3 4.3.5. Directional derivatives 5 4.3.6. Computing Directional Derivatives From Partial Derivatives 6 4.3.7. A nondiferentiable Function whose partial derivatives exist 11 4.3.8. Total Derivative 11 4. Univariate and Multivariate Differentiation (cont) 4.3. Multivariate calculus: functions from R n to R 4.3.1. Partial Derivative. . Cookbook approach: IF f : R n R , then f i ( · ) ∂f ( · ) ∂x i is the (usual) derivative oF the single variable Function you get by treating all other variables as constant. 1
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2 LECTURE #17: THU, OCT 25, 2007 PRINT DATE: AUGUST 21, 2007 (CALCULUS2) Example: f ( x ) = x 2 1 + 2 x 1 x 2 + x 2 2 = x 2 1 + 2 x 1 α + β f 1 ( x ) = 2 x 1 + 2 α + 0 = 2 x 1 + 2 x 2 A more graphical view of partial derivatives. Take a cross-section of the graph of the function along the i ’th axis, and look at the slope of the single-dimensional function you obtain in this way. This slope is the i ’th partial derivative. Similarly, you could take a diagonal cross-section, and get a diFerent one-dimensional slope. Gen- erally, you get what is called a directional derivative . Partial derivatives are just special kinds of directional derivatives: in particular, they tell you the slopes you obtain when you take cross- sections along the various axes. 4.3.2. The Gradient. Given a function f : R n R , the gradient of f is just the function from R n to R n that maps each x R n to the vector of partial derivatives of f at x . That is, f ( · ) = f 1 ( · ) . . . f i ( · ) . . . f n ( · ) Emphasize that f ( · ) is the exact analog for f : R n R . as f p ( · ) is for f : R R . In fact the words “slope of f,” “gradient of f” and “derivative of f” are synonomous.
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ARE211, Fall 2007 3 Emphasize the important distinction between the “gradient of f ” which is a function , written f or f ( · ) and the “gradient of f at x ,” which is a vector. 4.3.3. Crosspartial Derivative. A cross-partial derivative is just an entry in the matrix which is the derivative of the derivative. That is, take a function f : R n R . The i ’th partial derivative of this function f i ( · ) is a function, like any other, and if the function is diFerentiable, it has derivatives: f ij ( · ) 2 f ∂x i ∂x j ( · ) is the j ’th partial derivative of the function f i ( · ) ∂f ∂x i ( · ).
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mathCalculus2-07-draft - ARE211, Fall 2007 LECTURE #17:...

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