Math 21a: Multivariable calculus
Homework 1: Geometry and Distance
1 [Compare Stewart 9.1: 8] Find the distance from (6, 8, 10) to
each of the following:
a) The xy-plane, b) The yz-plane, c) the xz-plane, d) the x-axes,
e) the y-axes, f) the z-axes
Soluti
Math 21a: Multivariable calculus
Homework 4: Lines and Planes
1 [Compare Stewart 9.5: 14] a) Find a parametric equation for the
line L through the point (3, 4, 6) such that L is perpendicular to
the plane x 2y + 3z = 7.
b) Intersect the line L with the co
Math 21a: Multivariable calculus
Homework 6: Parametrized curves
.
1 [Compare Stewart 10.1:19-24 ]
2 [Compare Stewart 10.1: 38 ] Find a vector function that represents the curve of intersection of the two surfaces z = 4x2 + y 2
and the cubic cylinder y =
Math 21a: Multivariable calculus
Homework 7: Arc length and curvature
1 [Compare Stewart 10.3: 6] Find the length of the curve
r(t) = 24t, 16t3/2, 6t2 ,
where t [0, 5].
Solution:
We have r (t) = 12, 12 t, 6t and |r(t)| = 12 + 6t so that
5
0 12 + 6t dt = 1
Math 21a: Multivariable calculus
Homework 12: Partial derivatives
1 [Compare Stewart 11.3: 12 ] If f (x, y) = 4 x2 4y 2 , nd
fx(1, 0) and fy (1, 0) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.
Soluti
Math 21a: Multivariable calculus
Homework 8: Other coordinates
1 [Compare Stewart 9.7: 6/10] Change to cylindrical coordinates:
a) (2 3, 2, 1) b) (4, 3, 2)
Change to spherical coordinates: c) (0, 3, 1) d) (1, 1, 6)
Solution:
We compute:
(a) r = x2 + y 2 =
Math 21a: Multivariable calculus
Homework 11: Functions and Continuity
1 [Compare Stewart 11.1: 26] Draw a contour map of the following
function showing several level curves
f (x, y) = x2/(x3 + y 3 ) .
Try to draw some level curves without technology, the
Math 21a: Multivariable calculus
Homework 5: Functions of 2 and 3 variables
1 [Compare Stewart 9.6: 15,27]
Solution:
15) Overall, all 6 of these graphs have dierent traces in the
planes x = 0 and y = 0, so we can investigate these for each
function.
(a) f
Math 21a: Multivariable calculus
Fall 2013
Homework 3: Cross product
This homework is due Monday, 9/16 rsp Tuesday 9/17. This is the ocial homework sheet. The
problems are close to book problems but in general dierent.
1 [Compare Stewart 9.4: 10,12,16] Fi
Math 21a: Multivariable calculus
Homework 2: Vectors and Dot product
1 [Compare Stewart 9.2: 22/24]
a) Find a vector that has the same direction as 2, 5, 3 but has
length 6.
b) You pull a sled force F = 5, 1, 4 . The sled has velocity
v = 3, 1, 0 . The do