# 265L10 - LESSON 10 Directional Derivatives and the Gradient...

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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-→ ı + ∂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-→ ı + 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 we’ll 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-→ ı + ∂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, let’s 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 )-→ ı + f y (-→...
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265L10 - LESSON 10 Directional Derivatives and the Gradient...

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