Vector Calculus: Scalar Fields, Vector Fields, and Directional Derivatives, Lecture notes of Engineering Science and Technology

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MAT 247 Engineering Mathematics

Hakkı Ula¸s Unal¨ Dept. of Electrical-Electronics Eng. Eskisehir Technical University, Turkey

October 3, 2018

Today

(^1) Vector Calculus

Functions and Fields

If to each point a = [a 1 , a 2 , a 3 ] in some region of Cartesian

coordinate space that corresponds a scalar φ(a 1 , a 2 , a 3 ), then, φ(a)

is called a scalar field. And φ is a scalar function of the three

Cartesian coordinates.

Examples are: temperature of the ocean at position described by a or

A simple example is Euclidean distance of a point P = (x, y, z)

from a some fixed point Po = (x 0 , y 0 , z 0 ), which can be defined as

φ(P ) :=

(x − x 0 )^2 + (y − y 0 )^2 + (z − z 0 )^2

Functions and Fields

If to each point a = [a 1 , a 2 , a 3 ] in some region of Cartesian

coordinate space that corresponds a vector v(x 1 , x 2 , x 3 ), then, v(a)

is called a vector field, similarly, v(·) is called vector function.

Functions and Fields

If to each point a = [a 1 , a 2 , a 3 ] in some region of Cartesian

coordinate space that corresponds a vector v(x 1 , x 2 , x 3 ), then, v(a)

is called a vector field, similarly, v(·) is called vector function.

120130140 90 100110 (^20157080)

(^3025)

(^4035) 45

(^50)

(^2015) 10 50

Gravitational force field: the force of attraction of the earth (its

mass is M ) on a mass m at a point P = (x, y, z) is given by

F = −G

M m

r^3

r = −G

M m

r^3

(xi, yj, zk),

Some calculus definitions-convergence

Let {v(n)}n=1, 2 ,... be a sequence of vectors in R^3 , where

v(n)^ = (v

(n)

1 , v

(n)

2 ,... , v

(n)

m ). Then, it is said that^ {v(n)}n=1, 2 ,...

converges to a vector v = (v 1 , v 2 ,... , vm) if

lim

n→∞

|v(n)^ − v| = 0,

where v is called the limit of the sequence.

Some calculus definitions-convergence

Let v(t) be a vector function with variable t is said to converge to

a point to in R^3 as t approaches to some to if v(t) is defined in

some neighbourhood of to (often to is not in the neighbourhood)

and

lim

t→to

|v(t) − l| = 0.

where v is called the limit of the sequence.

Some calculus definitions-continuity

Let v(t) be a vector function with variable t and it is said to be

continuous at t = to if v(t) is defined in some neighbourhood of to

( the neighbourhood contains to ) and

lim

t→to

v(t) = v(to)

Rules for differentiation

Let u(t) and v(t) be vector functions in R^3 with variable t. Suppose that the vector functions are defined on a subset of R and they can be differentiable at some interior points of this subset. Then, (cv(t))′^ = cv(t)′

Rules for differentiation

Let u(t) and v(t) be vector functions in R^3 with variable t. Suppose that the vector functions are defined on a subset of R and they can be differentiable at some interior points of this subset. Then, (cv(t))′^ = cv(t)′ (u + v(t))′^ = u(t)′^ + v(t)′

Rules for differentiation

Let u(t) and v(t) be vector functions in R^3 with variable t. Suppose that the vector functions are defined on a subset of R and they can be differentiable at some interior points of this subset. Then, (cv(t))′^ = cv(t)′ (u + v(t))′^ = u(t)′^ + v(t)′ (u · v(t))′^ = u(t)′v(t) + u(t)v(t)′ (u × v(t))′^ = u(t)′^ × v(t) + u(t) × v(t)′

Geometric Interpretation of a Vector Function

Let

OP be a position vector, where O is the origin and P is the point r(t) = (x(t), y(t), z(t)). Then, P describes a curve while t varies in some range. Therefore, −−→ OP = r(t), where r(t) = (x(t), y(t), z(t)) is called the parametric equation of the curve described by P and t is the parameter that specifies the curve.

Example

Find the locus of −−→ P as t varies in 0 ≤ t ≤ 2 π if OP = (α cos(t), 0 , α sin(t))

Example

Let a and b be the position vectors relative to the origin of the points A and B. Show that the equation of the straight line through A to B can be expressed as r = a + (b − a)t,

Directional derivative

Some of the vector fields arising in applications, such as electrostatic fields, can be obtained from scalar fields.

Directional derivative

Some of the vector fields arising in applications, such as electrostatic fields, can be obtained from scalar fields. Let f (x, y, z) be a differentiable function in R^3 with Cartesian coordinates x, y, z. Then, the gradient of f (x, y, z) is defined as

grad f

gradf = ∇f =

[

∂f ∂x

∂f ∂y

∂f ∂z

]

∂f ∂x

i + ∂f ∂y

j + ∂f ∂z

k,

where the differential operator ∇ can be defined as

∂x i +

∂y j +

∂z k.