Introduction to
electromagnetism
Vector and Fields
What is Electromagnetism
Electromagnetism is a branch of physics involving the study of
the electromagnetic force, a type of physical interaction that
occurs between electrically charged particles. The
electromagnetic force is carried by electromagnetic fields
composed of electric fields and magnetic fields, and it is
responsible for electromagnetic radiation such as light. Wikipedia
Why Study Electromagnetism
Most everyday equipment involve electromagnetism
1. Mobile
2. Computers
3. Radio
4. Satellite communications
5. Lasers
6. Projectors
7. Light bulbs
………
Engineering Application by Electromagnetism
Antenna design
in cell phone
Photolithography in
semiconductor process
Electric motor
Vector and Fields
Scalars and Vectors
Scalars:
1. Requires only one number for its description
2. Example: Distance: 5m; Speed: 10 m/s; Temperature: 25oC, etc.
3. Algebra consists of addition, subtraction, multiplication, etc.
Vectors:
1. Requires more than one scalars for its description
2. Example: Velocity: 5km/s along the certain direction; 𝐴⃑ = 𝐴 𝑥⃑ + 𝐴 𝑦⃑ + 𝐴 𝑧⃑
3. The corresponding algebra is called Vector Algebra
Vector and Fields
Addition of two vectors:
 Place the tail of B the head of A
 Commutative: A+B=B+A
 Associative:(A+B)+C=A+(B+C)
Vector and Fields
Multiplication by a scalar:
 Multiplies the magnitude but leaves the direction unchanged.
 Distributive: C(A+B)=CA+CB
Dot product of two vector (scalar product):
 The dot product of two vectors is defined by A·B≡|A||B| cos θ ,
where θ is the angle they form when placed tail-to-tail.
 Commutative: A·B=B·A
 Distributive: A·(B+C)=A·B+A·C
Vector and Fields
Cross product of two vector (vector product):
 The cross product of two vectors is defined by A×B≡|A||B|sinθ 𝑛 ,
where 𝑛 is a unit vector pointing perpendicular to the plane of A and B.
 A hat is used to designate the unit vector and its direction is determined
by the right-hand rule.
 Not commutative: A×B=-B×A
 Distributive: A×(B+C)=A×B+A×C
Vector and Fields
Let 𝑥⃑, 𝑦⃑ and 𝑧⃑ are unit vectors parallel to the x, y, and z axes, respectively.
An arbitrary vector A can be expressed in terms of these basis vectors.
𝐴⃑ = 𝐴 𝑥⃑ + 𝐴 𝑦⃑ + 𝐴 𝑧⃑
The numbers Ax, Ay, and Az are called components.
Vector and Fields
Reformulate the four vector operations as a rule for
manipulating components:
To add vectors, add like components.
𝐴⃑ + 𝐵 = (𝐴 𝑥⃑ + 𝐴 𝑦⃑ + 𝐴 𝑧⃑) + (𝐵 𝑥⃑ + 𝐵 𝑦⃑ + 𝐵 𝑧⃑)
= (𝐴 +𝐵 )𝑥⃑ + (𝐴 +𝐵 )𝑦⃑ + (𝐴 +𝐵 )𝑧⃑)
To multiply by a scalar, multiply each component.
𝑎𝐴⃑ = 𝑎(𝐴 𝑥⃑ + 𝐴 𝑦⃑ + 𝐴 𝑧⃑) = 𝑎𝐴 𝑥⃑ + 𝑎𝐴 𝑦⃑ + 𝑎𝐴 𝑧⃑
Vector and Fields
To calculate the dot product, multiply like components, and add.
𝐴⃑ 𝐵 = (𝐴 𝑥⃑ + 𝐴 𝑦⃑ + 𝐴 𝑧⃑) (𝐵 𝑥⃑ + 𝐵 𝑦⃑ + 𝐵 𝑧⃑)
= 𝐴 𝐵 +𝐴 𝐵 +𝐴 𝐵
To calculate the cross product, form the determinant whose first
row are 𝑥⃑, 𝑦⃑ and 𝑧⃑, whose second row is A (in component form),
and whose third row is B.
𝑥⃑
𝐴⃑ × 𝐵 = 𝐴
𝐵
𝑦⃑
𝐴
𝐵
𝑧⃑
𝐴 = 𝐴 𝐵 − 𝐴 𝐵 𝑥⃑ +
𝐵
𝐴 𝐵 − 𝐵 𝐴 𝑦⃑ + (𝐴 𝐵 − 𝐴 𝐵 )𝑧⃑
Vector and Fields
Triple Products
Since the cross product of two vectors is itself a vector, it can be dotted or
crossed with a third vector to form a triple product.
 Scalar triple product: A·(B×C). Geometrically, |A·(B×C)| is the volume of a
parallelepiped generated by these three vectors as shown below.
Vector and Fields
Triple Products
 Vector triple product: A×(B×C). The vector triple product can
be simplified by the so-called BAC-CAB rule.
𝐴⃑ × (𝐵 × 𝐶⃑) = 𝐵 𝐴⃑ 𝐶⃑ − 𝐶⃑(𝐴⃑ 𝐵)
Note. 𝐴⃑ × (𝐵 × 𝐶⃑) ≠ (𝐴⃑ × 𝐵) × 𝐶⃑
𝐴⃑ × 𝐵 × 𝐶⃑ = −𝐶⃑ × 𝐴⃑ × 𝐵 = −(𝐴⃑ 𝐵 𝐶⃑ − 𝐵 𝐴⃑ 𝐶⃑ )
Vector and Fields
Position, Displacement and Separation Vectors
Position vector: The vector to that point from the origin.
𝑟⃑ = 𝑥𝑥⃑ + 𝑦𝑦⃑ + 𝑧𝑧⃑
Its magnitude (the distance from the origin)
𝑟=
𝑟⃑ 𝑟⃑ =
𝑥 +𝑦 +𝑧
Its direction unit vector (pointing radially outward)
𝑟̇⃑ =
𝑟⃑
𝑥𝑥⃑ + 𝑦𝑦⃑ + 𝑧𝑧⃑
=
𝑟
𝑥 +𝑦 +𝑧
The infinitesimal displacement vector from (x, y, z) to (x+dx, y+dy, z+dz)
𝑑𝑙⃑ = 𝑑𝑥𝑥⃑ + 𝑑𝑦𝑦⃑ + 𝑑𝑧𝑧⃑
Vector and Fields
Position, Displacement and Separation Vectors
In electrodynamics one frequently encounters problems
involving two points:
A source point r′, where an electric field is located A field
point r, at which you are calculating the electric field
𝛾⃑ = 𝑟 − 𝑟⃑, magnitude 𝛾⃑ = 𝑟 − 𝑟⃑
Unit vector
𝛾̇⃑ =
𝛾⃑
𝑟 − 𝑟⃑
=
𝛾
𝑟 − 𝑟⃑
Vector and Fields
Gradient
A theorem on partial derivatives states that:
𝜕𝐻
𝜕𝐻
𝜕𝐻
𝑑𝐻 =
𝑑𝑥 +
𝑑𝑦 +
𝑑𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝐻
𝜕𝐻
𝜕𝐻
=
𝑥⃑ +
𝑦⃑ +
𝑧⃑
𝑑𝑥𝑥⃑ + 𝑑𝑦𝑦⃑ + 𝑑𝑧𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
= 𝛻𝐻 𝑑𝑙⃑
The gradient of H is
𝜕𝐻
𝜕𝐻
𝜕𝐻
𝛻H =
𝑥⃑ +
𝑦⃑ +
𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
Vector and Fields
Gradient
 Geometrical interpretation: Like any vector, the gradient has
magnitude and direction.
 A dot product in abstract form is 𝑑𝐻 = 𝛻𝐻 𝑑𝑙⃑ = 𝛻𝐻 𝑑𝑙⃑ 𝑐𝑜𝑠𝜃
 The gradient ∇H points in the direction of maximum increase of
the function H.
 Analogous to the derivative of one variable, a vanishing derivative
signals a maximum, a minimum, or an inflection.
Vector and Fields
Gradient
Vector and Fields
The Operator 𝛻
 The gradient has the formal appearance of a vector, ∇, “multiplying”,
a scalar H
𝜕
𝜕
𝜕
𝛻H =
𝑥⃑ +
𝑦⃑ + 𝑧⃑ 𝐻
𝜕𝑥
𝜕𝑦
𝜕𝑧
 ∇ is a vector operator that acts upon H, not a vector that multiplies H
𝜕
𝜕
𝜕
𝛻=
𝑥⃑ +
𝑦⃑ + 𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
 ∇ mimics the behavior of an ordinary vector in virtually every way, if
we translate “multiply” by “act upon”
Vector and Fields
The Operator 𝛻
An ordinary vector A can be multiply in three ways:
 Multiply a scalar a : aA
 Multiply another vector (dot product): A·B
 Multiply another vector (cross product):A×B
The operator ∇ can act
 On a scalar function H: ∇H
 On a vector function v (dot product): ∇·v
 On a vector function v (cross product): ∇ × v
Vector and Fields
The Divergence
Divergence of a vector v is:
𝜕
𝜕
𝜕
𝛻 𝑣⃑ =
𝑥⃑ +
𝑦⃑ + 𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑣
𝜕𝑣
𝜕𝑣
=
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝑣 𝑥⃑ + 𝑣 𝑦⃑ + 𝑣 𝑧⃑
∇·v is a measure of how much the vector v spread out from the
point in question.
Vector and Fields
Vector and Fields
Curl of a vector v is:
𝑥⃑
𝜕
𝛻 × 𝑣⃑ =
𝜕𝑥
𝑣
𝑦⃑
𝜕
𝜕𝑦
𝑣
𝑧⃑
𝜕𝑣
𝜕𝑣
𝜕
=
−
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝜕𝑣
𝑥⃑ +
−
𝑦⃑ +
−
𝑧⃑
𝜕𝑧 𝜕𝑥𝑧
𝜕𝑥
𝜕𝑦
∇×v is a measure of how much the vector v curl around the point in question.
Vector and Fields
Product Rules
Vector and Fields
The sum rule:
𝑑
𝑑𝑓 𝑑𝑔
𝑓+𝑔 =
+
𝑑𝑥
𝑑𝑥 𝑑𝑥
𝛻
𝐴⃑ + 𝐵 = 𝛻 𝐴⃑ + 𝛻 𝐵
𝛻 𝑓 + 𝑔 = 𝛻𝑓 + 𝛻𝑔
𝛻 × 𝐴⃑ + 𝐵 = 𝛻 × 𝐴⃑ + 𝛻 × 𝐵
The rule for multiplying by a constant K:
𝛻
𝑑
𝑑𝑓
𝐾𝑓 = 𝐾
𝑑𝑥
𝑑𝑥
𝛻 𝐾𝑓 = 𝐾𝛻𝑓
𝐾𝐴⃑ = 𝐾𝛻 𝐴⃑
𝛻 × 𝐾𝐴⃑ = 𝐾𝛻 × 𝐴⃑
Vector and Fields
Product Rules
Scalar fg, vector fA
𝑑
𝑑𝑓
𝑑𝑔
𝑓𝑔 = 𝑔
+𝑓
𝑑𝑥
𝑑𝑥
𝑑𝑥
𝛻
𝛻 𝑓𝑔 = 𝑔𝛻𝑓 + 𝑓𝛻𝑔
𝑓𝐴⃑ = 𝑓𝛻 𝐴⃑ + 𝐴⃑ 𝛻𝑓 𝛻 × 𝑓𝐴⃑ = 𝑓𝛻 × 𝐴⃑ + 𝛻𝑓 × 𝐴⃑
Scalar 𝐴⃑ 𝐵, vector 𝐴⃑ × 𝐵
𝛻 𝐴⃑ 𝐵 = 𝐴⃑ × 𝛻 × 𝐵 + 𝐵 × 𝛻 × 𝐴⃑ + 𝐴⃑ 𝛻 𝐵 + (𝐵 𝛻)𝐴⃑
𝛻
𝐴⃑ × 𝐵 = 𝐵
𝛻 × 𝐴⃑ − 𝐴⃑
𝛻×𝐵
𝛻 × 𝐴⃑ × 𝐵 = 𝐵 𝛻 𝐴⃑ − 𝐴⃑ 𝛻 𝐵 + 𝐴⃑ 𝛻 𝐵 − 𝐵(𝛻 𝐴⃑)
Vector and Fields
Quotient Rules
Scalar f/g, vector A/g
𝑑 𝑓
=
𝑑𝑥 𝑔
𝑔
𝑑𝑓
𝑑𝑔
−𝑓
𝑑𝑥
𝑑𝑥
𝑔
𝑓
𝑔𝛻𝑓 − 𝑓𝛻𝑔
𝛻
=
𝑔
𝑔
𝐴⃑
𝑔𝛻 𝐴⃑ − 𝐴⃑ 𝛻𝑔
𝛻
=
𝑔
𝑔
𝐴⃑
𝑔𝛻 × 𝐴⃑ − 𝛻𝑔 × 𝐴⃑ 𝑔𝛻 × 𝐴⃑ + 𝐴⃑ × 𝛻𝑔
𝛻×
=
=
𝑔
𝑔
𝑔
Vector and Fields
Second Derivative
Scalar f, vector A
Five species of second derivatives.
 Divergence of gradient: 𝛻 𝛻𝑓
 Curl of gradient: 𝛻 × 𝛻𝑓
 Gradient of divergence: 𝛻(𝛻 𝐴⃑)
 Divergence of curl: 𝛻 (𝛻 × 𝐴⃑)
 Curl of curl: 𝛻 × (𝛻 × 𝐴⃑)
Vector and Fields
Second Derivative
Divergence of gradient
𝜕
𝜕
𝜕
𝜕𝑓
𝜕𝑓
𝜕𝑓
𝛻 𝛻𝑓 =
𝑥⃑ +
𝑦⃑ + 𝑧⃑
𝑥⃑ +
𝑦⃑ +
𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕 𝑓 𝜕 𝑓 𝜕 𝑓
--The Laplacian of f
=
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
Curl of gradient
𝑥⃑
𝜕
𝛻 × 𝛻𝑓 = 𝜕𝑥
𝜕𝑓
𝜕𝑥
𝑦⃑
𝜕
𝜕𝑦
𝜕𝑓
𝜕𝑦
𝑧⃑
𝜕
𝜕 𝜕𝑓 𝜕 𝜕𝑓
𝜕 𝜕𝑓
𝜕 𝜕𝑓
𝜕 𝜕𝑓
𝜕 𝜕𝑓
𝜕𝑧 =
−
𝑥⃑ +
−
𝑦⃑ +
−
𝑧⃑
𝜕𝑦 𝜕𝑧 𝜕𝑧 𝜕𝑦
𝜕𝑧 𝜕𝑥 𝜕𝑥 𝜕𝑧
𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥
𝜕𝑓
𝜕𝑧
= 0 Why?
Vector and Fields
Second Derivative
Gradient of divergence
𝜕𝐴
𝜕
𝜕
𝜕
𝜕𝐴
𝜕𝐴
𝑥⃑ +
𝑦⃑ + 𝑧⃑
+
+
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
=
+
+
𝑥⃑ +
+
+
𝑦⃑ +
+
+
𝜕𝑥
𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧
𝜕𝑦𝜕𝑥
𝜕𝑦
𝜕𝑦𝜕𝑧
𝜕𝑧𝜕𝑥 𝜕𝑧𝜕𝑦
𝜕𝑧
𝛻(𝛻 𝐴⃑) =
𝑧⃑
Divergence of curl
𝛻 (𝛻 × 𝐴⃑) =
𝜕
𝜕
𝜕
𝑥⃑ +
𝑦⃑ + 𝑧⃑
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝐴
𝜕𝐴
−
𝜕𝑦
𝜕𝑧
𝜕𝐴
𝜕𝐴
𝑥⃑ +
−
𝜕𝑧
𝜕𝑥
𝜕𝐴
𝜕𝐴
𝑦⃑ +
−
𝜕𝑥
𝜕𝑦
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
=
−
+
−
+
−
= 0 Why?
𝜕𝑥𝜕𝑦 𝜕𝑥𝜕𝑧
𝜕𝑦𝜕𝑧 𝜕𝑦𝜕𝑥
𝜕𝑧𝜕𝑥 𝜕𝑧𝜕𝑦
𝑧⃑
Vector and Fields
Second Derivative
Curl of curl
𝛻 × 𝛻 × 𝐴⃑ =
𝑥⃑
𝜕
𝜕𝑥
𝑦⃑
𝜕
𝜕𝑦
𝑧⃑
𝜕
𝜕𝑧
𝜕𝐴
𝜕𝐴
−
𝜕𝑦
𝜕𝑧
𝜕𝐴
𝜕𝐴
−
𝜕𝑧
𝜕𝑥
𝜕𝐴
𝜕𝐴
−
𝜕𝑥
𝜕𝑦
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
−
−
+
𝑥⃑ +
−
−
+
𝑦⃑ +
−
−
+
𝑧⃑
𝜕𝑦𝜕𝑥
𝜕 𝑦
𝜕 𝑧
𝜕𝑧𝜕𝑥
𝜕𝑧𝜕𝑦
𝜕 𝑧
𝜕 𝑥
𝜕𝑥𝜕𝑦
𝜕𝑥𝜕𝑧
𝜕 𝑥
𝜕 𝑦 𝜕𝑦𝜕𝑧
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
=
+
+
−(
+
+
) 𝑥⃑ +
+
+
−(
+
+
) 𝑦⃑
𝜕𝑥
𝜕𝑦𝜕𝑥 𝜕𝑧𝜕𝑥
𝜕𝑥
𝜕 𝑦
𝜕 𝑧
𝜕𝑦𝜕𝑥
𝜕𝑦
𝜕𝑦𝜕𝑧
𝜕𝑥
𝜕 𝑦
𝜕 𝑧
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
𝜕 𝐴
+
+
+
−(
+
+
) 𝑧⃑ = 𝛻 𝛻 𝐴⃑ − 𝛻 𝐴⃑
𝜕𝑧𝜕𝑥 𝜕𝑧𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕 𝑦
𝜕 𝑧
=
𝐴⃑ × (𝐵 × 𝐶⃑) = 𝐵 𝐴⃑ 𝐶⃑ − 𝐶⃑(𝐴⃑ 𝐵)
Integral Calculus
Vector and Fields
Line integrals: an integral is an expression of the form
𝑣⃑ 𝑑𝑙⃑
Where v is a vector function, dl is the infinitesimal displacement
vector, and the integral is to be carried out along a prescribed
path P from point a to point b.
 Put a circle on the integral, in the path in question forms a
closed loop
𝑣⃑ 𝑑𝑙⃑
Vector and Fields
Integral Calculus (Line)
The line integral is independent of the path, and is determined
entirely by the end points, e. g.
𝑊=
𝐹⃑ 𝑑𝑙⃑
Conservative
Vector and Fields
Vector and Fields
Vector and Fields
Integral Calculus (Surface)
 Surface integrals: an integral is an expression of the form
𝑣⃑ 𝑑𝑎⃑
where v is a vector function, and da is the infinitesimal patch of area, with
direction perpendicular to the surface.
 The value of a surface integral depends on the particular surface chosen,
but there is a special class of vector functions for which it is independent
of the surface, and is determined entirely by the boundary.
Vector and Fields
Vector and Fields
Integral Calculus (Volume)
 Volume integrals: an integral is an expression of the form
𝑇𝑑𝜏
where T is a scalar function, and d τ is an infinitesimal
volume element. In Cartesian coordinates, dτ =dxdydz
 The volume integrals of vector functions
𝑉𝑑𝜏 =
𝑉 𝑑𝜏𝑥⃑ +
𝑉 𝑑𝜏𝑦⃑ +
𝑉 𝑑𝜏𝑧⃑
Vector and Fields
Vector and Fields
Fundamental theorem of calculus:
𝑑𝑓
𝑑𝑥 =
𝑑𝑥
𝑑𝑓 = 𝑓 𝑏 − 𝑓(𝑎)
Geometrical Interpretation: two ways to determine the total
change in the function:
 Go step-by-step adding up all the tiny increments as you go
 Subtract the values at the ends
Vector and Fields
Fundamental theorem of gradient:
 A scalar function of three variables T(x, y, z) changes by a small
amount.
𝑑𝑇 = (𝛻𝑇) 𝑑𝑙⃑
 The total change in T in going from a to b along the path selected is
(𝛻𝑇) 𝑑𝑙⃑ = 𝑇 𝑏 − 𝑇(𝑎)
Geometrical Interpretation
Measure the high of each floor and add them all up
Place an altimeter at the top and the bottom, subtract the readings at the ends
Vector and Fields
Fundamental theorem of gradient:
∫ (𝛻𝑇) 𝑑𝑙⃑ = 𝑇 𝑏 − 𝑇(𝑎) the right side of this equation
makes no reference to the path---only to the end points. Thus
gradients have special property that their line integrals are path
independent.
Corollary 1: ∫ (𝛻𝑇) 𝑑𝑙⃑ is independent of path taken from a to b.
Corollary 2: ∮(𝛻𝑇) 𝑑𝑙⃑ = 0, since the beginning and end points are
identical, and hence T(b)-T(a)=0.
Vector and Fields
Potential Energy and Conservative Forces
 Potential energy defined in terms of work done by the
associated conservative force.
𝑈 −𝑈 =−
𝐹⃑ 𝑑𝑙⃑
 Conservative forces tend to minimize the potential energy
within any system: It allowed to, an apple falls to the
ground and a spring returns to its natural length.
 Non-conservative force does not imply it is dissipative, for
example, magnetic force, and also does not mean it will
decrease the potential energy, such as hand force.
Vector and Fields
Conservative and Non-conservative Forces
 The distinction between conservative and non-conservative forces
is best stated as follows
 A conservative force may be associated with a scalar potential
energy function, whereas a non-conservative force cannot.
𝑈 −𝑈 =−
𝐹⃑ 𝑑𝑙⃑
𝐹⃑ = −𝛻𝑈
Vector and Fields
Conservative and Non-conservative Forces
 A conservative force can be derived from a scalar potential energy
function
𝐹⃑ = −𝛻𝑈
 The negative sign indicates that the force points in the direction
of decreasing potential energy.
Gravity Ug = mgy, F = -mg
Spring Ug = ½(kx2), F = -kx
Vector and Fields
The Fundamental Theorem for Divergences
 The fundamental theorem for divergences states that:
∫
𝛻 𝑣⃑ 𝑑𝜏 = ∮ 𝑣⃑ 𝑑𝑎⃑
 The integration of a derivative (in this case the divergence) over a region (in
this case a volume) is equal to the value of the function at the boundary (in
this case the surface that bounds the volume)
 This theorem has at least three special names: Gauss’s theorem, Green’s
theorem, or the divergence theorem.
Vector and Fields
Vector and Fields
The Fundamental Theorem for Curls
The fundamental theorem for curls---Stokes theorem--states that:
𝛻 × 𝑣⃑
𝑑𝑎⃑ =
𝑣⃑ 𝑑𝑙⃑
The integration of a derivative (here, the curl) over a region (here, a patch of
surface) is equal to the value of the function at the boundary (in this case the
perimeter of the patch)
Vector and Fields
The Fundamental Theorem for Curls
Ambiguity in Stokes theorem: Concerning the boundary line integral,
which way are we supposed to go around (clockwise or
counterclockwise)? The right-hand rule.
Corollary 1: ∫ 𝛻 × 𝑣⃑ 𝑑𝑎⃑ depends only on the boundary lines,
not on the particular surface used.
Corollary 2: ∮ 𝛻 × 𝑣⃑ 𝑑𝑎⃑ for any closed surface, since the
boundary line shrinks down to a point
These corollaries are analogous to those for the gradient theorem.
Vector and Fields
Vector and Fields
Curvilinear Coordinates (Spherical Polar Coordinates)
The spherical (polar) coordinates (r, θ, φ)of a point Pare defined below:
r: the distance from the origin (the magnitude of the position vector)
θ: the angle down from the z-axis (called polar angle)
φ: The angle around from the x-axis (call the azimuthal angle).
𝑥 = 𝑟𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙
𝑦 = 𝑟𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙
𝑧 = 𝑟𝑐𝑜𝑠𝜃
Vector and Fields
Curvilinear Coordinates (Spherical Polar Coordinates)
The direction of the coordinates: the unit vector 𝑟⃑, 𝜃⃑, 𝜙
They constitute an orthogonal (mutually perpendicular) basis set (just like 𝑥⃑, 𝑦⃑, 𝑧⃑)
So any vector A can be expressed in terms of them:
𝐴⃑ = 𝐴 𝑟⃑ + 𝐴 𝜃⃑ + 𝐴 𝜙
𝑟⃑ = 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙𝑥⃑ + 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙𝑦⃑ + 𝑐𝑜𝑠𝜃𝑧⃑
𝜃⃑ = 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙𝑥⃑ + 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙𝑦⃑ − 𝑠𝑖𝑛𝜃𝑧⃑
𝜙 = −𝑠𝑖𝑛𝜙𝑥⃑ + 𝑐𝑜𝑠𝜙𝑦⃑
In terms of Cartesian unit vector
Vector and Fields
Curvilinear Coordinates (Spherical Polar Coordinates)
 𝑟⃑, 𝜃, 𝜙 are associated with particular point P, and they
change direction as P moves around. For example, 𝑟⃑ always
points radially outward, but “radially outward” can be the x
direction, the y direction, or any other direction, depending
on where you are.
 Since the unit vectors are function of position, we must
handle the differential and integral with care.
1. Differentiate a vector that is expressed in spherical coordinates.
2. Do not take 𝑟⃑, 𝜃, 𝜙 outside an integral.
Vector and Fields
Curvilinear Coordinates (Spherical Polar Coordinates)
𝑑𝑙⃑ = 𝑑𝑙 𝑟⃑ + 𝑑𝑙 𝜃⃑ + 𝑑𝑙 𝜙 = 𝑑𝑟𝑟⃑ + 𝑟𝑑𝜃𝜃⃑ + 𝑟𝑠𝑖𝑛𝜃𝑑𝜙𝜙
𝑑𝑎⃑ = 𝑑𝑙 𝑑𝑙 𝑟⃑ = 𝑟 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙𝑟⃑
𝑑𝜏 = 𝑑𝑙 𝑑𝑙 𝑑𝑙 = 𝑟 𝑠𝑖𝑛𝜃𝑑𝑟𝑑𝜃𝑑𝜙
Vector and Fields
Curvilinear Coordinates (Spherical Polar Coordinates)
The vector derivatives in spherical coordinates:
Vector and Fields
Curvilinear Coordinates (Cylindrical Coordinates)
The cylindrical coordinates (s, φ , z)of a point Pare defined below:
s: the distance from the z axis
φ: the same meaning as in spherical coordinates
z: the same as Cartesian
𝑥 = 𝑠𝑐𝑜𝑠𝜙
𝑦 = 𝑠𝑠𝑖𝑛𝜙
𝑧=𝑧
𝑟⃑ = 𝑐𝑜𝑠𝜙𝑥⃑ + 𝑠𝑖𝑛𝜙𝑦⃑
𝜙 = −𝑠𝑖𝑛𝜙𝑥⃑ + 𝑐𝑜𝑠𝜙𝑦⃑
𝑧⃑ = 𝑧⃑
In terms of Cartesian unit vector
Vector and Fields
Curvilinear Coordinates (Cylindrical Coordinates)
𝑑𝑙⃑ = 𝑑𝑙 𝑟⃑ + 𝑑𝑙 𝜙 + 𝑑𝑧𝑧⃑ = 𝑑𝑠𝑠⃑ + 𝑠𝑑𝜙𝜙 + 𝑑𝑧𝑧⃑
𝑑𝜏 = 𝑑𝑙 𝑑𝑙 𝑑𝑙 = 𝑠𝑑𝑠𝑑𝜙𝑑𝑧
Vector and Fields
Curvilinear Coordinates (Cylindrical Coordinates)
Vector and Fields
The Dirac Delta Function
A vector function 𝑣⃑ = ⃑
1 𝜕
1
1 𝜕
The divergence of this vector function is 𝛻 𝑣⃑ =
(𝑟
)=
(1) = 0
𝑟 𝜕𝑟
𝑟
𝑟 𝜕𝑟
The surface integral of this function is
𝛻 𝑣⃑ 𝑑𝜏 =
𝑣⃑ 𝑑𝑎⃑ =
1
𝑟 𝑠𝑖𝑛𝜃 𝑑𝜙𝑑𝜃 = 2𝜋
𝑟
Dirac delta function
𝑠𝑖𝑛𝜃𝑑𝜃 ≠ 0
Vector and Fields
The 1D Dirac Delta Function
The 1-D Dirac delta function can be pictured as an infinitely high,
infinitesimally narrow “spike”, with area just 1.
0 𝑖𝑓 𝑥 ≠ 0
𝛿 𝑥 =
with
∞ 𝑖𝑓 𝑥 = 0
𝛿 𝑥 𝑑𝑥 = 1
Technically, δ(x) is not a function at all, since its value is not finite at x=0.
Such function is called the generalized function, or distribution.
Vector and Fields
The 1D Dirac Delta Function
If f(x) is continuous function, then the product f(x)δ(x) is zero everywhere
except at x=0. It follows that f(x)δ(x)=f(0)δ(x). In particular,
𝑓 𝑥 𝛿 𝑥 𝑑𝑥 = 𝑓 0
𝛿 𝑥 𝑑𝑥 = 𝑓(0)
We can shift the spike from x=0 to x=a
0 𝑖𝑓 𝑥 ≠ 𝑎
𝛿 𝑥−𝑎 =
with
∞ 𝑖𝑓 𝑥 = 𝑎
𝑓 𝑥 𝛿 𝑥 − 𝑎 𝑑𝑥 = 𝑓 𝑎
𝛿 𝑥 − 𝑎 𝑑𝑥 = 1
𝛿 𝑥 − 𝑎 𝑑𝑥 = 𝑓(𝑎)
Vector and Fields
The 1D Dirac Delta Function
𝑥 𝛿 𝑥 − 2 𝑑𝑥 = 8
𝑥 𝛿 𝑥 − 2 𝑑𝑥 = 0
Although δ(x) is not a legitimate function, integrals over δ(x) are perfectly
acceptable. It is best to think of the delta function as something that is always
intended for use under an integral sign.
In particular, two expressions involving delta function are considered equal if:
𝑓 𝑥 𝐷 (𝑥)𝑑𝑥 =
𝑓(𝑥)𝐷 (𝑥)𝑑𝑥
Vector and Fields
The 1D Dirac Delta Function
1
𝛿 𝑘𝑥 =
𝛿(𝑥) where k is any (nonzero) constant
𝑘
Changing variables, we let y = kx, so that x = y/k, and dx = (1/k)dy. If k
is positive, the integration still runs from -∞ to ∞, but if k is negative,
then x = ∞ implies y = - ∞, and vice versa, so the order of the limits is
reversed. Restoring the "proper" order costs a minus sign. Thus
1
𝑓 𝑥 𝛿 𝑘𝑥 𝑑𝑥 =
𝑘
𝑓
𝑦
1
1
𝑘 𝛿 𝑦 𝑑𝑦 = ± 𝑘 𝑓 0 = 𝑘 𝑓(0)
1
𝛿 𝑘𝑥 =
𝛿(𝑥)
𝑘
Vector and Fields
The 3D Dirac Delta Function
𝛿 𝑟 = 𝛿(𝑥)𝛿(𝑦)𝛿(𝑧)
Its volume integral is
𝛿 𝑟 𝑑τ =
𝛿 𝑥 𝛿 𝑦 𝛿 𝑧 𝑑𝑥𝑑𝑦𝑑𝑧 = 1
As in the 1-D case, the integral with delta function picks out the value of
the function at the location of the spike
𝑓 𝑟 𝛿 𝑟 − 𝑎 𝑑τ = 𝑓(𝑎)
Vector and Fields
The 3D Dirac Delta Function
We found that the divergence of ⃑
is zero everywhere except at the origin,
and yet its integral over any volume containing the origin is a constant of 4π.
The Dirac delta function can be defined as:
𝑟⃑
𝛻
= 4𝜋𝛿 𝑟
𝑟
More generally:
𝛻
𝛾⃑
𝛻
= 4𝜋𝛿 𝛾
𝛾
1
=𝛻
𝛾
1
𝛻
𝛾
=𝛻
𝛾⃑ = 𝑟⃑ − 𝑟′
𝛾⃑
−
𝛾
= −4𝜋𝛿 𝛾
Vector and Fields
The Helmholtz Theorem
To what extent is a vector function F determined by its divergence and curl?
The divergence of F is a specified scalar function D,
𝛻 𝐹⃑ = 𝐷
and the curl of F is a specified vector function C,
𝛻 × 𝐹⃑ = 𝐶⃑ with 𝛻
𝛻 × 𝐹⃑ = 𝛻 𝐶⃑ = 0
Helmholtz theorem guarantees that the field F is uniquely determined by the
divergence and curl with appropriate boundary conditions.
Vector and Fields
The Helmholtz Theorem (Potential)
If the curl of a vector field (F) vanishes (everywhere), then F can be
written as the gradient of a scalar potential (V):
𝛻 × 𝐹⃑ = 0
⇒
𝐹⃑ = −𝛻𝑉
If the divergence of a vector field (F) vanishes (everywhere), then F can
be expressed as the curl of a vector potential (A):
𝛻 𝐹⃑ = 0
⇒
𝐹⃑ = 𝛻 × 𝐴⃑