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Unformatted text preview: LESSON 10 Directional Derivatives and the Gradient READ: Section 15.5 NOTES: There is a certain vector formed from the partial derivatives of a function z = f ( x,y ) that pops up in a lot of applications. This vector is called the gradient of f , and the definition is f ( x,y ) = f x- + f y- . By the way, if you collect odd words to use in Scrabble or crossword puzzles, the symbol is called nabla . f is read grad f or gradient f or del f , the last because nabla is an upside down delta. As an example, if f ( x,y ) = x sin y , then f ( x,y ) = sin y- + x cos y- . It is important to keep in mind that f is a vector quantity. Strictly speaking, the correct way to write the gradient is- f ( x,y ) since the gradient is a vector quantity, and many texts do write it correctly . But this text does not even use bold font for the , so well follow the convention used here and omit the arrow above the , and make the extra effort to remember f is a vector quantity. The definition of the gradient is extended in the natural way to functions of any number of variables. So if w = f ( x,y,z ), then the gradient of f is f ( x,y,z ) = f x- + f y- + f z- k . Perhaps the neatest way to write the formula for the gradient of f is to think of the points in f s domain as vectors. In other words, lets write- x = ( x,y ) if the domain of f is in the plane or- x = ( x,y,z ) if the domain of f is part of 3-space. Then we can write the gradient of f as f (- x ) = f x (- x )- + f y (-...
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