1.2 Problems
21
Extract the square root of both members, separate the variables, and integrate
again, introducing the second arbitrary constant C 2 .
Complex variables
Complex number z = r (cos θ + i sin θ), where i =
√ −1
z
n
= cos nθ + i sin nθ
Analytic functions
A function f of the complex variable z is analytic at a point z o if its derivative f
(z)
exists not only at z o but at every point z in some neighborhood of z o . As an example
if f (z) =
1
z
then f
(z) = −
1
z 2 (z = 0). Thus f is analytic at every point except the
point z = 0, where it is not continuous, so that f
(0) cannot exist. The point z = 0
is called a singular point.
Contour
A contour is a continuous chain of finite number of smooth arcs. If the contour
is closed and does not intersect itself, it is called a closed contour. Boundaries of
triangles and rectangles are examples. Any closed contour separates the plane into
two domains each of which have the points of C as their only boundary points. One
of these domains is called the interior of C, is bounded; the other, the exterior of C,
is unbounded.
Contour integral is similar to the line integral except that here one deals with the
complex plane.
The Cauchy integral formula
Let f be analytic everywhere within and on a closed contour C. If z o is any point
interior to C, then
f (z o ) =
1
2πi
C
f (z) dz
z − z o
where the integral is taken in the positive sense around C.
1.2 Problems
1.2.1 Vector Calculus
1.1 If φ =
1
r
, where r = (x
2
+ y
2
+ z
2 )
1/2 , show that ∇φ =
r
r 3 .
1.2 Find a unit vector normal to the surface x y
2
+ xz = 1 at the point (−1, 1, 1).
1.3 Show that the divergence of the Coulomb or gravitational force is zero.
1.4 If A and B are irrotational, prove that A×B is Solenoidal that is div ( A×B) =
0
21
Extract the square root of both members, separate the variables, and integrate
again, introducing the second arbitrary constant C 2 .
Complex variables
Complex number z = r (cos θ + i sin θ), where i =
√ −1
z
n
= cos nθ + i sin nθ
Analytic functions
A function f of the complex variable z is analytic at a point z o if its derivative f
(z)
exists not only at z o but at every point z in some neighborhood of z o . As an example
if f (z) =
1
z
then f
(z) = −
1
z 2 (z = 0). Thus f is analytic at every point except the
point z = 0, where it is not continuous, so that f
(0) cannot exist. The point z = 0
is called a singular point.
Contour
A contour is a continuous chain of finite number of smooth arcs. If the contour
is closed and does not intersect itself, it is called a closed contour. Boundaries of
triangles and rectangles are examples. Any closed contour separates the plane into
two domains each of which have the points of C as their only boundary points. One
of these domains is called the interior of C, is bounded; the other, the exterior of C,
is unbounded.
Contour integral is similar to the line integral except that here one deals with the
complex plane.
The Cauchy integral formula
Let f be analytic everywhere within and on a closed contour C. If z o is any point
interior to C, then
f (z o ) =
1
2πi
C
f (z) dz
z − z o
where the integral is taken in the positive sense around C.
1.2 Problems
1.2.1 Vector Calculus
1.1 If φ =
1
r
, where r = (x
2
+ y
2
+ z
2 )
1/2 , show that ∇φ =
r
r 3 .
1.2 Find a unit vector normal to the surface x y
2
+ xz = 1 at the point (−1, 1, 1).
1.3 Show that the divergence of the Coulomb or gravitational force is zero.
1.4 If A and B are irrotational, prove that A×B is Solenoidal that is div ( A×B) =
0