Engineering Calculus Notes 254

Engineering Calculus Notes 254 - direction of Fow on the...

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242 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION The graph of this function is an elliptic paraboloid opening down; that is, it can be viewed as a hill whose peak is above the origin, at height f (0 , 0) = 49. The gradient of this function is −→ f ( x,y ) = ( 2 x ) −→ ı + ( 6 x ) −→ ; at the point (4 , 1), −→ f (4 , 1) = 8 −→ ı 6 −→ has length v v v −→ f (4 , 1) v v v = r 8 2 + 6 2 = 10 and the unit vector parallel to −→ f (4 , 1) is −→ u = 4 5 −→ ı 3 5 −→  . This means that at the point 4 units east and one unit north of the peak, a climber who wishes to gain height as fast as possible should move in the direction given on the map by −→ u ; by moving in this direction, the climber will be ascending at 10 units per unit of horizontal motion from an initial height of f (4 , 1) = 30. Alternatively, if a stream Fowing down the mountain passes the point 4 units east and one unit north of the peak, its
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Unformatted text preview: direction of Fow on the map will be in the opposite direction, the direction of steepest descent . The analogue of Remark 3.3.5 for a function of three variables holds for the same reasons. Note that in either case, the gradient lives in the domain of the function; thus, although the graph of a function of two variables is a surface in space, its gradient vector at any point is a vector in the plane. Chain Rules or two dierentiable real-valued functions of a (single) real variable, the Chain Rule tells us that the derivative of the composition is the product of the derivatives of the two functions: d dt V V V V t = t [ f g ] = f ( g ( t )) g ( t ) ....
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