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Unformatted text preview: MECH201 Engineering Analysis (2009) p. 50 8 PARTIAL DIFFERENTIATION A REVISON Consider an analytic function which has two depend variables: x and y . That is z = f(x,y). Figure 8.1 A 3D profile. If z is the vertical axis, the xaxis is in the EastWest direction and yaxis in the NorthSouth direction, then the partial differentiation of z with respect to x while holding y constant: y z x , or simply z x is the slope of the surface represented by z = f(x,y) in the EW direction (along the xaxis). Similarly z y represents the slope in the NSouth direction. A B C y x z = f(x,y) ( f/ x)dx ( f/ y)dy MECH201 Engineering Analysis (2009) p. 51 The increment of height (that is increment of z) for a small increase x in the EW direction (from point A to C as shown in the Figure) is given by ( ) EW z z x x = . Further, the increase in height from C to E, that is moving North (holding x constant) is given by: ( ) NS z z y y = . Thus the total increase in height z after moving from A to C and then to E is: z z z x y x y = + For infinitesimal increment, we write d for and the above may be replaced by: z z dz dx dy x y = + If one travels from A to E along the direction which has a bearing of (measured anticlockwise from the East), the increment of height is given by the same formula, namely: except that we must have ( ) tan y x = . Let the projection of the path AE on the horizontal plane (the xyplane) is s . the average slope of the path from A to E is given by: tan AE z z x z y s x s y s = = + Taking the limit of s b 0 we arrived at a formula for the total differentiation of z: dz z dx z dy ds x ds y ds = + note that: cos and sin dx dy ds ds = = . In general for z = f(x 1 , x 2 , x 3 , x 4 , x 5 , x i ,) z z z x y x y = + MECH201 Engineering Analysis (2009) p. 52 we have i i i z dz dx x = and the slope is given by: i i i dx dz z ds x ds = MECH201 Engineering Analysis (2009) p. 53 9 THE ELLIPTIC PARTIAL DIFFERENTIAL EQUATION 9.1 Steady State Heat Conduction Section 29.1 Here we will develop a PDE for the computation of temperature distribution on a 2D. The distribution of temperature is represented by the isotherms lines of constant temperature. A 2dimensional plate as shown below has a prescribed temperature on its sides: C on side ABCD and 100 C on side EF. The temperature varies linearly from C to 100 C on sides AF and DE. Figure 9.1 A 2D conduction domain As this is a steady state situation, the net heat flow into an element of the plate (size x by y ) is zero at all time....
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 Three '09
 WeihuaLi

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