FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

1

APPLICATION OF PARTIAL DERIVATIVES

( PARTIAL DERIVATIVE in Engineering Problems)

1. MATERIALS Castigliano’s theorem says the displacement, , under a force P is given by :

PU

where U is the internal strain energy of the body. For the

following, find :

a) AELPU

2

2

b) EILPU

969 32

2. STRUCTURES Given the stress function, 2442),( yBxyAxyx where A and B are

constants, find the stresses, x and y where 2

2

yx

and

2

2

xy

.

3. FLUID MECHANICS

The continuity equation is defined as : 0

yv

xu

, where vu, are

velocities of the fluid in the x, y directions respectively. Show that 22 xyu , xyv 2 will satisfy the continuity equation.

FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

2

4. THERMODYNAMICS

Given the ideal gas equation, RTPV where P is pressure, V is

volume, T is temperature and R is the gas constant. Given isothermal

compressibility

PV

Vk 1 and

TV

V1

.

Show that : P

k 1 and

T1

.

5. Suppose that 22 yxD is the length of the diagonal of a rectangle whose sides have lengths x and y that are allowed to vary. a) Find a formula for the rate of change of D with respect to x if varies with y held constant.

b) Hence, use this formula to find the rate of change of D with respect to x at the point when 3x and 4y .

6. Let 32 5),( yyxyxf a) Find the slope of the surface ),( yxfz in the x-direction at the point (1,-2). b) Find the slope of the surface ),( yxfz in the y-direction at the point (1,-2).

7. A point moves along the intersection of the elliptic paraboloid 22 3yxz and the plane 1y . At what rate is z changing with x when the point is at )7,1,2( ?

FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

3

8. The volume V of a right circular cylinder is given by the formula hrV 2 , where r is the radius and h is the height.

a) Find a formula for the instantaneous rate of change of V with respect to r if r change and h remains constant.

b) Find a formula for the instantaneous rate of change of V with respect to h if h change and r remains constant.

c) Suppose that h has a constant value of 4 inches, but r varies. Find

the rate of change of V with respect to r at the point where r =6 inches.

d) Suppose that r has a constant value of 8 inches, but h varies. Find the rate of change of V with respect to h at the point where h =10 inches.

9. Show that the functions given below are solutions of the wave equation

2

22

2

2

xWc

tW

a) ctxtcxW 22cossin b) tcxW 22ln

c) tcxW 22tan

10. Show that the functions given below satisfy the

three-dimensional Laplace equation : 02

2

2

2

2

2

zf

yf

xf or

two-dimensional Laplace equation : 02

2

2

2

yf

xf

a) 222 2zyxf b) zyxzf 223. 32

c) xef y 2cos2 d) 21222 zyxf

FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

4

11. Determine whether xyxyxu 2),( is a solution of the

partial differential equation : yyu

xu

2

12. Verify that )sin(xyz satisfies the equation :

0)( 222

2

2

2

zyx

yz

xz

13. Determine whether yxxyyxu ),( is a solution of the partial

differential equation : xyx

uxy

uy 222

2

2

14. Given 22),( yxyxz , verify that ),( yxz is a solution of :

2122

2

2

2

2

)(

yx

yz

xz

FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

5

APPLY SMALL CHANGES TO ENGINEERING APPLICATIONS

15. If RVI and V = 250 volts , R = 50 ohms. Find the change in I resulting

from an increase in 1 volt in V and an increase of 0.5 ohm in R.

16. If 4

3

dwsy , find the percentage change in y when w increase by 2% ,

s decrease by 3% and d decrease by 1%.

17. The deflection y at the centre of a circular plate suspended at the edge and

uniformly loaded is given by 3

4

tkwdy , where w = total load, d diameter

of plate, t thickness and k is a constant. Calculate the approximate

change in y if w is increased by 3%, d is decreased by 2.5% and t is

increased by 4%.

18. The two sides forming the right angle of a right-angled triangle are denoted by

a and b. The hypotenuse is h. If there are possible errors of %5.0 in

measuring a and b, find the maximum possible error in calculating:

a) the area of the triangle, A

b) the length of h

b

a h

FKB 14202: TUTORIAL 3 Partial Derivatives

There is no failure except in no longer trying - Elbert Hubbard

6

ANSWERS

1. a) AEPL b) AE

PL163 3

2. 224

2

2

122 yBxAyxy

422

2

2

212 BxyAxyx

5 a) 22 yx

xxD

b)

53

6. a) -4 b) 61 7) 4 8 a) rh2 b) 2r c) 80.15048 d) 06.20164 15) I reduces 0.03 ampere. 16) y is decreased by 3% 17) y decrease by 19% approximately. 18) a) AofA %1 b) hofh %5.0