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lecture_9 (dragged) 3

# lecture_9 (dragged) 3 - y − f x y ∆ y ∆ y ∆ t Upon...

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4 MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 Total Di ff erentials. Say that z = f ( x, y ) is a function. We will show that dz = f x dx + f y dy. This is called the total di ff erential of the function z = f ( x, y ). First we make sense of this notation. Say that we have a path γ = γ ( t ) in the plane R 2 . That is, given any t R , we have a point ( x, y ) = γ ( t ) R 2 . Then the composition z ( t ) = f γ ( t ) is actually a function of t . Choose a “small” di ff erence t , and define the di ff erences x and y by ( x, y ) = γ ( t + t ) γ ( t ) i.e., ( x + x, y + y ) = γ ( t + t ) . Figure 4. Plot of γ ( t ) vs. t -4.8 -4 -3.2 -2.4 -1.6 -0.8 0 0.8 1.6 2.4 3.2 4 4.8 -3.2 -2.4 -1.6 -0.8 0.8 1.6 2.4 3.2 We have the ratio z ( t + t ) z ( t ) t = f γ ( t + t ) f γ ( t ) t = f ( x + x, y + y f ( x, y ) t = f ( x + x, y + y ) f ( x, y + y ) t + f ( x, y + y ) f ( x, y ) t = f ( x + x, y + y ) f ( x, y + y ) x x t + f ( x, y + y ) f ( x, y ) y y
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Unformatted text preview: y ) − f ( x, y ) ∆ y ∆ y ∆ t . Upon taking limits, we ²nd that dz dt = lim ∆ t → ² f ( x + ∆ x, y ) − f ( x, y ) ∆ x ∆ x ∆ t + f ( x, y + ∆ y ) − f ( x, y ) ∆ y ∆ y ∆ t ³ = ∂f ∂x dx dt + ∂f ∂y dy dt . This is called the total derivative of the function z = f ( x, y ). Note that it involves partial derivatives . This formula is the same as the more familiar result involving the dot product of the gradient: dz dt = ∇ ( f ◦ γ ) · γ ° ( t ) in terms of ∇ f ( x, y ) = ´ ∂f ∂x , ∂f ∂y µ . Upon multiplying both sides by the di±erential dt we ²nd the total di±erential above....
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