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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