201-Lect 11 - MECH201 Engineering Analysis (2009) p. 50 8...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 3-D profile. If z is the vertical axis, the x-axis is in the East-West direction and y-axis in the North-South 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 E-W direction (along the x-axis). Similarly z y represents the slope in the N-South 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 E-W 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 xy-plane) 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 2-dimensional 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 2-D 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....
View Full Document

Page1 / 13

201-Lect 11 - MECH201 Engineering Analysis (2009) p. 50 8...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online