lecture_9 (dragged) 3 - y ) f ( x, y ) y y t . Upon taking...

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4 MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 2 Total Diferentials. 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 diferential 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±erence ∆ t , and de²ne the di±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
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Unformatted text preview: y ) f ( x, y ) y y t . Upon taking limits, we nd that dz dt = lim t f ( x + x, y ) f ( x, y ) x x t + f ( x, y + y ) f ( x, y ) y y t = 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 ) ( t ) in terms of f ( x, y ) = f x , f y . Upon multiplying both sides by the dierential dt we nd the total dierential above....
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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