Worksheet 12: 16.3
MATH 23:
Fall Semester 2014
1. Explain why the following is true: Whenever the line integral of a vector eld around every
closed curve is zero, the line integral along a curve with xed endpoints has a constant value
independent of the p
Worksheet 13: 16.4, 16.5
MATH 23:
Spring Semester 2015
1. Use Greens theorem to calculate the integral of F around the curve, oriented counterclockwise. You might want to check your answer by parametrizing the path and computing the
line integral.
(i) F =
Worksheet 15: 16.7
MATH 23:
Spring Semester 2015
1. Consider the part of the surface z = 4 x2 y 2 (in meters) above the circle in the xy
plane of radius 1m centered at the origin, oriented with its normal pointing up, that is above
z = 0. What is the ux o
Worksheet 11: 16.1, 16.2
MATH 23:
Fall Semester 2014
1. Sketch the vector elds in the xy-plane.
(i) F (x, y) = 2i + 3j
(ii) F (r) = r/ r
(iii) F (x, y) = y i + xj
(iv) F (x, y) = (x + y)i + (x y)j
2. Write a formula for the two-dimensional vector eld with
Worksheet 7: 15.1,15.2
MATH 23:
Spring Semester 2015
1. Sketch the region of integration and evaluate the iterated integrals
3
4
(i)
3
(4x + 3y) dx dy
0
2
(ii)
0
6xy dy dx
0
0
2. Find the volume of the region bounded by z = x2 , 0 x 5 and the planes y = 0
Worksheet 10: 15.7, 15.8, 15.9
MATH 23:
1. Sketch the region of integration.
1
1
(i)
0
1
1x2
1
1x2
f (x, y, z) dz dx dy
(ii)
0
1z 2
1z 2
Spring Semester 2015
1y 2 z 2
f (x, y, z) dx dy dz
1y 2 z 2
2. Find the volume of the region bounded by the surfaces
Worksheet 8: 15.2, 15.3, 15.4
MATH 23
Spring Semester 2015
1. Sketch the region of integration and evaluate the iterated integrals
3
4
(i)
3
(4x + 3y) dx dy
0
2
(ii)
0
6xy dy dx
0
0
2. Find the volume of the region bounded by z = x2 , 0 x 5 and the planes
Worksheet 5: 14.1, 14.3
MATH 23:
Spring Semester 2015
2
1. Plot vertical traces of the function z = ex sin y by setting:
(i) x = 0, x = 1, x = 1.
(ii) y = 0, y = /2, y = /2.
Sketch the graph of the function.
2. Sketch a contour map for the function with a
Worksheet 6: 14.4, 14.5, 14.6
MATH 23:
Spring Semester 2015
1. For the following functions, nd an equation of the tangent plane at the given point.
(i) f (x, y) = 1 (x3 + 8y 3 ) at the point (2, 1, 8)
2
(ii) f (x, y) = xey/x at the point (1, 1, e)
(iii)f
Worksheet 4: 13.1, 13.2, 13.3, 13.4
MATH 23:
Fall Semester 2014
1. Parameterize the following curves:
(i) y = esin(x) in the xy-plane.
(ii) z = y 3 + y + 1 in the yz-plane.
(iii) x =
1
z 2 +1
in the xz-plane.
2. Parametrize the line going through (1, 2, 0
MATH 23: Vector Calculus
Worksheet 1: 12.12
Fall Semester 2014
1. Which two of the three points P1 (1, 2, 3), P2 (3, 2, 1) and P3 (1, 1, 0) are closest to each other?
2. Describe the set of points whose distance from the x-axis is 2. Describe the set of p
MATH 23: Vector Calculus
Worksheet 3: 12.5, 12.6
Spring Semester 2015
1. Find an equation of the plane through (2, 3, 2) and parallel to the plane 3x + y + z = 4.
2. Find the points where the plane z = 5x 4y + 3 intersects each of the coordinate axes. Fin
MATH 23: Multi-variable Calculus
Worksheet 2: 12.34
Spring Semester 2014
1. Write = 3 i + 2 j 6 k as the sum of two vectors, one parallel, and one perpendicular,
a
to d = 2 i 4 j + k
2. Consider a point P and the plane through the point Po with normal
Worksheet 14: 16.6
MATH 23:
Spring Semester 2015
1. Find a parametrisation representing a bagel if the bagel is obtained by rotating around the
z-axis a small circle of radius 1 in the xz-plane so that the center of the small circle traces a
circle of rad