Introduction to Engineering Mathematics



Vector Calculus for Engineers

CME100, Fall 2004

Handout #2

Functions of Several Variables, Partial Derivatives, Chain Rule

1. Find the limit: [pic] by examining the following cases:

a) approaching (0,0) along [pic]

b) approaching (0,0) along [pic]

c) approaching (0,0) along [pic]

2. For [pic] find [pic], [pic], [pic], [pic], [pic], [pic] and [pic].

>> syms x y z

>> diff('x*y^3*sin(y*z)',x)

y^3*sin(y*z)

>> diff('x*y^3*sin(y*z)',y)

3*x*y^2*sin(y*z)+x*y^3*cos(y*z)*z

>> diff('x*y^3*sin(y*z)',z)

x*y^4*cos(y*z)

3. For the function [pic] determine [pic] and [pic] in polar coordinates [pic] and [pic].

4. For [pic] such that [pic] determine a) [pic] b) [pic]

Linearization, Differential

5. Two resistors [pic] and [pic] are connected in parallel. If [pic] and [pic] and if these values may be in error by as much as [pic], estimate the maximum error in the calculated value of the combined resistance.

Gradient, Directional Derivative

6. For the following function [pic] determine:

a) gradient of z at (1,1)

b) directional derivative in the direction of [pic]at (1,1)

c) directions of maximum increase, maximum decrease, and no change in z

d) a particle moves by a distance of 0.1 units in the direction of the gradient. Determine the approximate change in z

e) a vector normal to the surface [pic] at (1,1)

meshc makes a surface plot with level curves for a function in 3D

>> [x,y]=meshgrid(-2:0.1:2);

>> z=10-x.^2-y.^2;

>> meshc(x,y,z)

Maxima and Minima, Lagrange Multipliers

7. Find the least amount of plywood needed to construct a closed rectangular box of a given volume V.

Function fminsearch minimizes functions

of several variables

>> fminsearch('2*x(1)*x(2)+2/x(1)+2/x(2)',[1,2])

ans =

1.0000 1.0000

8. Determine an equation of a line that best fits n data points [pic] in the least squares sense. Solve for the slope and the intercept for the following set of data points:

|n |x |y |

|1 |0 |0 |

|2 |1 |2 |

|3 |2 |1 |

|4 |3 |2 |

Function polyfit determines best fit (linear and nonlinear) to a given set of data

>> x=[0 1 2 3];

>> y=[0 2 1 2];

>> polyfit(x,y,1)

ans =

0.5000 0.5000

9. The temperature over a semicircular disk of radius 1 is given by: [pic]. Find all absolute maxima and minima (if any) over the specified region.

10. Redo Problem 7 using the method of Lagrange Multipliers.

-----------------------

z

y

[pic]

[pic]

[pic]

[pic]

x

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