M2_3.1_RM

3.1

Introduction to Vectors, Gradient and Directional Derivative

Introduction

The underlying elements in vector analysis are vectors and scalars. We

use the notation ℝ to denote the real line which is identified with the set

of real numbers, ℝ2 to denote the Cartesian plane, ℝ3 to denote the

ordinary 3-space. We denote vectors by bold face Roman letters.

Vectors

There are qualities in physics and science characterized by both

magnitude and direction, such as displacement, velocity, force, and

acceleration. To describe such quantities, we introduce the concept of a

vector as a directed line segment 𝑃𝑄 from one point 𝑃 to another point

𝑄. Here P is called the initial point or origin of 𝑃𝑄 , and Q is called the

terminal point, end, or terminus of the vector.

We will denote vectors by bold-faces letters or letters with an arrow

over them. Thus the vector 𝑃𝑄 may be denoted by 𝑨 or 𝐴 as in following

figure. The magnitude or length of the vector is then denoted by

𝑃𝑄 , 𝐴 , 𝐴 , 𝑜𝑟 𝐴.

The following comments apply.

(a) Two vectors 𝑨 and 𝑩 are equal if they have the same magnitude

and direction regardless of their initial point. Thus 𝑨 = 𝑩 in

following 1st figure.

(b) A vector having direction opposite to that of a given vector 𝑨 but

having the same magnitude is denoted by – 𝑨 [see 2nd following

figure] and is called the negative of 𝑨.

Scalars

Other quantities in physics and science are characterized by

magnitude only, such as mass, length, and temperature. Such quantities

are often called scalars to distinguish them from vectors. However, it must

be emphasized that apart from units, such as feet, degrees, etc., scalars are

nothing more than real numbers. Thus we can denote them, as usual, by

ordinary letters. Also, the real numbers 0 and 1 are part of our set of

scalars.

Vector Algebra

There are two basic operations with vectors:

(a) Vector Addition; (b) Scalar Multiplication.

𝑨 𝑨

𝑩 -A

(a) Vector Addition

Consider vectors 𝑨 and 𝑩, pictured in Fig(a). The sum or resultant of 𝑨

and 𝑩, is a vector 𝑪 formed by placing the initial point of 𝑩 on the

terminal point of 𝑨 and them joining the initial point of 𝑨 to the terminal

point of 𝑩, pictured in fig(b). The sum 𝑪 is written 𝑪 = 𝑨 + 𝑩. This

definition here is equivalent to the Parallelogram Law for vector addition,

pictured in fig(c).

Extensions to sums of more than two vectors are immediate. Consider,

for example, vectors 𝑨, 𝑩, 𝑪, 𝑫 in fig(a). Then fig(b) shows how to obtain

the sum of resultant E of the vectors 𝑨, 𝑩, 𝑪, 𝑫, that is, by connecting the

end of each vector to the beginning of the next vector.

𝑩

𝑨

(𝑎) (𝑏)

𝑨

𝑩

C=A+B

(𝑐)

𝑨

𝑩

(𝑎)

𝑨

𝑪

𝑩

𝑫

(𝑏)

𝑨

𝑩

𝑫

𝑪

𝑬 = 𝑨 + 𝑩 + 𝑪 + 𝑫

The difference of vectors 𝑨 and 𝑩, denoted by 𝑨 − 𝑩, is that vector 𝑪,

which added to 𝑩, gives 𝑨. equivalently, 𝑨 − 𝑩, may be defined as

𝑨 + −𝑩 .

If 𝑨 = 𝑩, then 𝑨 − 𝑩 is defined as the null or zero vector; it is

represented by the symbol 𝟎. It has zero magnitude and its direction is

undefined. A vector that is not null is a proper vector. All vectors will be

assumed to be proper unless otherwise stated.

(b) Scalar Multiplication

Multiplication of a vector 𝑨 by a scalar 𝑚 produces a vector 𝑚𝑨 with

magnitude 𝑚 times the magnitude of 𝑨 and the direction of 𝑚𝑨 is in

the same or opposite of 𝑨 according as m is positive or negative. If

𝑚 = 0, then 𝑚𝑨 = 0, the null vector.

Laws of Vector Algebra

The following theorem applies

Theorem: Suppose 𝑨, 𝑩, 𝑪 are vectors and 𝑚 and 𝑛 are scalars. Then the

following laws hold:

𝐴1 𝑨 + 𝑩 + 𝑪 = 𝑨 + 𝑩 + 𝑪 Associative Law for

Addition

𝐴2 There exists a zero vector 𝟎 such that, for every vector𝐴.

𝑨 + 𝟎 = 𝟎 + 𝑨 = 𝑨 Existence of Zero Element

𝐴3 For ever vector 𝑨, there exists a vector – 𝑨 such that

𝑨 + −𝑨 = −𝑨 + 𝑨 = 𝟎 Existence of Negatives

𝐴4 𝑨 + 𝑩 = 𝑩 + 𝑨 Commutative Law for

Addition

𝑀1 𝑚(𝑨 + 𝑩) = 𝑚𝑨 + 𝑚𝑩 Distributive Law

𝑀2 𝑚 + 𝑛 𝑨 = 𝑚𝑨 + 𝑛𝑨 Distributive Law

𝑀3 𝑚 𝑛𝑨 = (𝑚𝑛)𝑨 Associative Law

𝑀4 1 𝑨 = 𝑨 Unit Multiplication

The above eight laws are the axioms that define an abstract structure

called a Vector space.

The above laws split into two sets, as indicated by their labels. The first

four laws refer to vector addition. One can then prove the following

properties of vector addition.

a) Any sum 𝑨𝟏 + 𝑨𝟐 + ⋯ + 𝑨𝒏 of vectors requires no parentheses

and does not depend on the order of the summands.

b) The zero vector 𝟎 is unique and the negative – 𝑨 of a vector 𝑨 is

unique

c) ( Cancellation Law) If 𝑨 + 𝑪 = 𝑩 + 𝑪, then 𝑨 = 𝑩.

The remaining four laws refer to scalar multiplication. Using these

additional laws, we can prove the following properties.

PROPOSITION:

a) For any scalar 𝑚 and zero vector 𝟎, we have 𝑚𝟎 = 𝟎.

b) For any vector 𝑨 and scalar 0, we have 0𝑨 = 0.

c) If 𝑚𝑨 = 0, then 𝑚 = 0 or 𝑨 = 𝟎.

d) For any vector 𝑨 and scalar 𝑚, we have −𝑚 𝑨 = 𝑚 −𝑨 =

− 𝑚𝑨

Unit vectors

Unit vectors are vectors having unit length. Suppose A is any vector with

length 𝑨 > 0. Then 𝑨/ 𝑨 is a unit vector, denoted by a, which has the

same directions as A. Also, any very A may be represented by a unit

vectors a in the direction of A multiplied by the magnitude of A .That is,

𝑨 = 𝑨 𝑎.

Example: Suppose 𝑨 = 3. Then 𝒂 = 𝑨 /3 is a unit vector in the

directions of A. Also , 𝑨 = 3𝒂.

Rectangular Unit Vectors i, j, k

An important set of unit vectors, denoted by i, j and k, are those having

the directions, respectively, of the positive 𝑥, 𝑦, 𝑎𝑛𝑑 𝑧 axes of three-

dimensional rectangular coordinate system. The coordinate system shown

in fig, which we use unless otherwise stated, is called a right handed

coordinate system. The system is characterized by the following property.

If we curl the fingers of the right hand in the direction of a 900rotation

from the positive 𝑥 − 𝑎𝑥𝑖𝑠 to the positive 𝑦 − 𝑎𝑥𝑖𝑠, then the thumb will

point in the direction of the positive 𝑧 − 𝑎𝑥𝑖𝑠.

Generally speaking, suppose nonzero vectors A,B,C have the same initial

point and are not coplanar. Then A,B,C are said to form a right-hand

system or dextral system if a right-threaded screw rotated through an angle

less than 1800 from A to B will advance in the direction C as shown in

fig.

Components of a vector

Any vector A in three dimensions can be represented with an initial point

at the origin 0 = 0,0,0 and its end point at some point, say,

(𝐴1 , 𝐴2 , 𝐴3), then the vectors, 𝐴1𝒊, 𝐴2𝒋, 𝐴3𝒌 are called the component

vectors of A in the 𝑥, 𝑦, 𝑧 directions, and the scalars 𝐴1 , 𝐴2 , 𝐴3 are called

the components of A in the x, y, z directions , respectively

The sum of 𝐴1𝒊, 𝐴2𝒋, 𝑎𝑛𝑑 𝐴3𝒌 is the vector A , so we may write

𝑨 = 𝐴1𝒊 + 𝐴2𝒋 + 𝐴3𝒌

𝑥

𝑦

𝑧

𝒌 𝒋

𝒊 𝑜

(𝑎)

𝑨 𝑩

𝑪

(𝑏)

𝑜

𝑥

𝑧

𝑦

𝑨

𝐴3𝒌

𝐴2𝒋

𝐴1𝒊

(𝑐)

The magnitude of A follows

𝑨 = 𝐴12 + 𝐴2

2 + 𝐴32

Consider a point 𝑃 𝑥, 𝑦, 𝑧 in space. The vector r form the origin 0 to the

point 𝑃 is called the position vector (or radius vector). Thurs r may be

written

𝒓 = 𝑥𝒊 + 𝑦𝒋 + 𝑧𝒌

It has magnitude 𝒓 = 𝑥2 + 𝑦2 + 𝑍2

The following proposition applies,

Proposition: Suppose 𝑨 = 𝐴1𝒊 + 𝐴2𝒋 + 𝐴3𝒌 and 𝑩 = 𝐵1𝒊 + 𝐵2𝒋 + 𝐵3𝒌.

Then

i. 𝑨 + 𝑩 = 𝐴1 + 𝐵1 𝒊 + 𝐴2 + 𝐵2 𝒋 + 𝐴3 + 𝐵3 𝒌

ii. 𝑚𝑨 = 𝑚 𝐴1𝒊 + 𝐴2𝒋 + 𝐴3𝒌 = 𝑚𝐴1 𝒊 + 𝑚𝐴2 𝒋 + 𝑚𝐴3 𝒌

Example: Suppose 𝑨 = 3𝑖 + 5𝑗 − 2𝑘 and 𝑩 = 4𝑖 − 8𝑗 + 7𝑘

a. To find 𝑨 + 𝑩 and corresponding components , obtaining 𝑨 + 𝑩 =

7𝒊 − 3𝒋 + 5𝒌

b. To find 3𝑨 − 2𝑩, first multiply by the scalars and then add:

3𝑨 − 2𝑩 = 9𝒊 + 15𝒋 − 6𝒌 + −9𝒊 + 16𝒋 − 14𝒌

= 𝒊 + 31𝒋 − 20𝒌

c. To find 𝑨 and 𝑩 , take the square root of the sum of the square

of the components:

𝑨 = 9 + 25 + 4 = 38 and 𝑩 = 16 + 64 + 49 = 129

The DOT and CROSS Product

Definition: Let 𝒂 and 𝒃 be two vectors. The scalar product or dot product

of 𝒂 and 𝒃 is defined to be 𝒂. 𝒃 = 𝑎𝑏cos𝜃 where 𝜃 is the angle between

the two vectors when drawn from a common origin.

Note:

𝒂. 𝒃 = 𝒃. 𝒂.

𝒂. 𝒂 = 𝒂 2 = 𝑎2.

𝒂. 𝒃 = 0 if 𝑎 and 𝑏 are perpendicular vectors.

𝒂. 𝒃 + 𝒄 = 𝒂. 𝒃 + 𝒂. 𝒄.

𝒊. 𝒊 = 𝒋. 𝒋 = 𝒌. 𝒌 = 1.

𝒊. 𝒋 = 𝒋. 𝒌 = 𝒌. 𝒊 = 0.

If 𝒂 = 𝑎1𝑖 + 𝑎2𝑗 + 𝑎3𝑘 and 𝒃 = 𝑏1𝑖 + 𝑏2𝑗 + 𝑏3𝑘 then 𝒂. 𝒃 =

𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3.

Definition: Let 𝒂, 𝒃 be two nonzero vectors. Then the vector product or

cross product of 𝒂 and 𝒃 is a vector perpendicular to both 𝒂 and 𝒃 with

magnitude 𝑎𝑏 sin 𝜃 where 0 ≤ 𝜃 ≤ 𝜋 is the angle between 𝒂 and 𝒃 and

whose direction is along a unit vector 𝒏 such that 𝒂, 𝒃, 𝒏 from a right

handed system. Thus 𝒂 × 𝒃 = 𝑎𝑏 sin 𝜃 𝒏

Note:

𝒂 × 𝒃 = Area of the parallelogram with 𝑎 and 𝑏 as adjacent sides.

𝒂 × 𝒃 = −(𝒃 × 𝒂).

𝒂 × 𝒃 = 0 if 𝒂 and 𝒃 are parallel.

𝒂 × 𝒃 + 𝒄 = 𝒂 × 𝒃 + 𝒂 × 𝒄.

𝒊 × 𝒊 = 𝒋 × 𝒋 = 𝒌 × 𝒌 = 0.

𝒊 × 𝒋 = 𝒌 = −𝒋 × 𝒊, 𝒋 × 𝒌 = 𝒊 = −𝒌 × 𝒋, 𝒌 × 𝒊 = 𝒋 = −𝒊 × 𝒌.

If 𝒂 = 𝑎1𝒊 + 𝑎2𝒋 + 𝑎3𝒌 and 𝒃 = 𝑏1𝒊 + 𝑏2𝒋 + 𝑏3𝒌 then 𝒂 × 𝒃 =

𝑎2𝑏3 − 𝑎3𝑏2 𝒊 + 𝑎3𝑏1 − 𝑎1𝑏3 𝒋 + 𝑎1𝑏2 − 𝑎2𝑏1 𝒌.

= 𝒊 𝒋 𝒌𝑎1 𝑎2 𝑎3

𝑏1 𝑏2 𝑏3

.

Definition: The scalar triple product or box product of three vectors

𝒂, 𝒃, 𝒄 is defined to be the scalar 𝒂. 𝒃 × 𝒄 . It is sometimes denoted by

𝒂 𝒃 𝒄 .

𝒂. 𝒃 × 𝒄 = 𝒂 𝒃 𝒄 =

𝑎1 𝑎2 𝑎3

𝑏1 𝑏2 𝑏3

𝑐1 𝑐2 𝑐3

.

Note:

𝒂. 𝒃 × 𝒄 =Volume of the parallelepiped turned by the

coterminous edges 𝒂, 𝒃, 𝒄.

𝒂 𝒃 𝒄 = 𝒃 𝒄 𝒂 = 𝒄 𝒂 𝒃 .

𝒂 𝒃 𝒄 = − 𝒃 𝒂 𝒄 = − 𝒄 𝒃 𝒂 = − 𝒂 𝒄 𝒃 .

The vectors 𝒂, 𝒃, 𝒄 are coplanar if any only if 𝒂 𝒃 𝒄 = 0.

𝒂 × 𝒃 × 𝒄 = 𝒂. 𝒄 𝒃 − 𝒂. 𝒃 𝒄.

𝒂 × 𝒃 × 𝒄 = 𝒂. 𝒄 𝒃 − 𝒃. 𝒄 𝒂.

𝒂 × 𝒃 . 𝒄 × 𝒅 = 𝒂. 𝒄 𝒂. 𝒅𝒃. 𝒄 𝒃. 𝒅

.

𝒂 × 𝒃 × 𝒄 × 𝒅 = 𝒂 𝒃 𝒅 𝒄 − 𝒂 𝒃 𝒄 𝒅.

Scalar and vector fields:

Scalar filed: A scalar function 𝐹(𝑥, 𝑦, 𝑧) defined over some region of

space 𝐷 is a function that assigns to each point 𝑃0 in 𝐷 with coordinates

𝑥0 , 𝑦0 , 𝑧0 the number 𝐹 𝑃0 = 𝐹 𝑥0 , 𝑦0 , 𝑧0 . The set of all numbers

𝐹(𝑃) for all points 𝑃 in 𝐷 are said to form a scalar field over 𝐷.

Example:

The temperature at any point within or on the earth’s surface at a

certain time defines a scalar field.

The scalar function of position 𝐹 𝑥, 𝑦, 𝑧 = 𝑥𝑦𝑧2 for (𝑥, 𝑦, 𝑧)

inside the unit sphere 𝑥2 + 𝑦2 + 𝑧2 = 1 defines a scalar field

throughout the unit sphere.

Vector field: More general than a scalar field 𝐹(𝑥, 𝑦, 𝑧) is a vector field

defined by a vector function 𝑭(𝑥, 𝑦, 𝑧) over some region of space 𝐷 that

assigns to each point 𝑃0 in 𝐷 with coordinates 𝑥0 , 𝑦0 , 𝑧0 the vector

𝑭 𝑃0 = 𝑭 𝑥0 , 𝑦0 , 𝑧0 with its tail at 𝑃0.Functions of this type are called

either vector functions or vector-valued functions.

Example:

The vector-valued function 𝑭 𝑥, 𝑦, 𝑧 = 𝑥 − 𝑦 𝒊 − (𝑦 − 𝑧)𝒋 +

𝑥𝑦𝑧 − 2 𝒌, for (𝑥, 𝑦, 𝑧) inside the ellipsoid 𝑥2

𝑎2 +𝑦2

𝑏2 +𝑧2

𝑐2 = 1,

defines a vector field throughout the ellipsoid.

Suppose the velocity at any point within a moving fluid is known at a

certain time defines a vector field.

Limits and continuity of vector functions of a single real variable

A vector function of a single real variable 𝑭 𝑡 = 𝑓1 𝑡 𝒊 + 𝑓2 𝑡 𝒋 +

𝑓3 𝑡 𝒌 is said to have 𝒍 as its limit at 𝑡0, written lim𝑡→𝑡0

𝑭 𝑡 = 𝒍, where 𝒍 =

𝑙1𝒊 + 𝑙2𝒋 + 𝑙3𝒌.If lim 𝑡→𝑡0

𝑓1 𝑡 = 𝑙1 , lim𝑡→𝑡0

𝑓2 𝑡 = 𝑙2 , and lim𝑡→𝑡0

𝑓3 𝑡 = 𝑙3. If,

in addition, the vector function is defined at 𝑡0 and lim 𝑡→𝑡0

𝑭 𝑡 = 𝑭(𝑡0),

then 𝑭(𝑡) is said to be continuous at 𝑡0 . A vector function 𝑭(𝑡) that is

continuous for each 𝑡 in the interval 𝑎 ≤ 𝑡 ≤ 𝑏 is said to be continuous

over the interval. A vector function of a single real variable that is not

continuous at a point 𝑡0 is said to be discontinuous at 𝑡0 .

Differentiability and the derivative of a vector function of a single real

variable

The vector function of a single real variable 𝑭 𝑡 = 𝑓1 𝑡 𝒊 + 𝑓2 𝑡 𝒋 +

𝑓3 𝑡 𝒌 defined over the interval 𝑎 ≤ 𝑡 ≤ 𝑏 and 𝑡0𝜖 𝑎 𝑏 . Then

lim𝑡→𝑡0

𝑭 𝑡 −𝑭(𝑡0)

𝑡−𝑡0, it exists, is called the derivative of 𝑭(𝑡) at 𝑡0 and is denoted

by 𝑭′ 𝑡0 or 𝑑𝑭

𝑑𝑡 at 𝑡 = 𝑡0 . We also say that 𝑭 𝑡 is differentiable at a

point 𝑡0 in the interval if its components are differentiable at 𝑡0. It is said

to be differentiable over the interval if it is differentiable at each point of

the interval, and when 𝑭 𝑡 is differentiable its derivative with respect to 𝑡

is

𝑑𝑭

𝑑𝑡=

𝑑𝑓1

𝑑𝑡𝒊 +

𝑑𝑓2

𝑑𝑡𝒋 +

𝑑𝑓3

𝑑𝑡𝒌.

Note: Every differentiable function is continuous.

When 𝑑𝑭

𝑑𝑡 is differentiable, the second order derivative 𝑑2𝑭/𝑑𝑡2 is

defined as 𝑑2𝑭

𝑑𝑡2 =𝑑

𝑑𝑡

𝑑𝑭

𝑑𝑡 and, in general, provided the derivates exist,

𝑑𝑛 𝑭

𝑑𝑡𝑛 =𝑑

𝑑𝑡

𝑑𝑛−1𝑭

𝑑𝑡𝑛−1 , 𝑓𝑜𝑟 𝑛 ≥ 2.

Properties:

Let 𝒖 𝑡 and 𝒗 𝑡 be differentiable function of 𝑡 over some interval

𝑎 ≤ 𝑡 ≤ 𝑏.

𝑑𝐶

𝑑𝑡= 0 ( C is constant vector).

𝑑

𝑑𝑡 𝜆𝒖 = 𝜆

𝑑𝒖

𝑑𝑡 ( an arbitrary constant scalar).

𝑑

𝑑𝑡 𝒖 ± 𝒗 =

𝑑𝒖

𝑑𝑡±

𝑑𝒗

𝑑𝑡.

𝑑

𝑑𝑡 𝒖. 𝒗 =

𝑑𝒖

𝑑𝑡. 𝒗 + 𝒖.

𝑑𝒗

𝑑𝑡.

𝑑

𝑑𝑡 𝒖 × 𝒗 =

𝑑𝒖

𝑑𝑑𝑡× 𝒗 + 𝒖 ×

𝑑𝒗

𝑑𝑡.

It 𝒖 𝑡 is a differentiable function of 𝑡 and 𝑡 = 𝑡 𝑠 is a

differentiable function of 𝑠, then

𝑑𝒖

𝑑𝑠=

𝑑𝒖

𝑑𝑡.𝑑𝑡

𝑑𝑠.

Partial derivatives of a vector function of real variables

Let 𝑭 be a vector function of scalar variable 𝑝,q, 𝑡. Then we write 𝑭 =

𝑭 𝑝, 𝑞, 𝑡 . Treating 𝑡 as a variable and 𝑝, 𝑞 as constants, we define

lim𝛿𝑡→0𝑭 𝑝 ,𝑞 ,𝑡+𝛿𝑡 −𝑭 𝑝 ,𝑞 ,𝑡

𝛿𝑡 if exists, is called partial derivative of 𝑭 with

respect to 𝑡 and is donated by 𝜕𝑭

𝜕𝑡.

Similarly, we can define 𝜕𝑭

𝜕𝑝,𝜕𝑭

𝜕𝑞 also.

It 𝜕𝑭

𝜕𝑡 exists

𝜕2𝑭

𝜕𝑡2 and 𝜕2𝑭

𝜕𝑝𝜕𝑡 are defined as

𝜕2𝑭

𝜕𝑡2 =𝑑

𝑑𝑡

𝜕𝑭

𝜕𝑡 ,

𝜕2𝑭

𝜕𝑝𝜕𝑡=

𝜕

𝜕𝑝

𝜕𝑭

𝜕𝑡 etc.

Properties:

𝜕

𝜕𝑡 𝜙𝒖 =

𝜕𝜙

𝜕𝑡𝒖 + 𝜙

𝜕𝒖

𝜕𝑡 (𝜙 is a scalar differential function ).

𝜕

𝜕𝑡 𝜆𝒖 = 𝜆

𝜕𝒖

𝜕𝑡 .

𝜕

𝜕𝑡 𝒖 ± 𝒗 =

𝜕𝒖

𝜕𝑡±

𝜕𝒗

𝜕𝑡.

𝜕

𝜕𝑡 𝒖. 𝒗 =

𝜕𝒖

𝜕𝑡. 𝒗 + 𝒖.

𝜕𝒗

𝜕𝑡.

𝜕

𝜕𝑡 𝒖 × 𝒗 =

𝜕𝒖

𝜕𝑑𝑡× 𝒗 + 𝒖 ×

𝜕𝒗

𝜕𝑡.

Let 𝑭 = 𝐹1𝒊 + 𝐹2𝒋 + 𝐹3𝒌, where 𝐹1 , 𝐹2 , 𝐹3 are differential scalar

functions of more than one variable, then

𝜕𝑭

𝜕𝑡=

𝜕𝐹1

𝜕𝑡 𝒊 +

𝜕𝐹2

𝜕𝑡 𝒋 +

𝜕𝐹3

𝜕𝑡 𝒌.

Level surfaces:

Let 𝑓 𝑥, 𝑦, 𝑧 be a single valued continuous scalar function defined at

every point 𝑝 ∈ 𝐷. Then 𝑓 𝑥, 𝑦, 𝑧 = 𝑐 (constant), defines the equation

of a surface and is called a level surface of the function. For different

values of 𝑐, we obtain different surface, no two of which intersect. For

example, if 𝑓 𝑥, 𝑦, 𝑧 represent temperature in a medium, then

𝑓 𝑥, 𝑦, 𝑧 = 𝑐 represents a surface on which the temperature is a

constant 𝑐. Such surfaces are called isothermal surfaces.

Example:

Find the level surface of the scalar fields in space, defined by 𝑓 𝑥, 𝑦, 𝑧 =

𝑧 − 𝑥2 + 𝑦2.

Solution: We find that 𝑓 𝑥, 𝑦, 𝑧 = 𝑐

⇒ 𝑧 − 𝑥2 + 𝑦2 = 𝑐 ⇒ 𝑥2 + 𝑦2 = 𝑧 − 𝑐 2

The level surfaces are cones.

Directional Derivatives and the Gradient Operator

Consider a scalar function 𝑤 = 𝑓 𝑥, 𝑦, 𝑧 with continuous first order

partial derivatives with respect to 𝑥, 𝑦, and 𝑧 that is defined in same region

D of Space, and let a space curve Γ in D have the parametric equations

𝑥 = 𝑥 𝑡 , 𝑦 = 𝑦 𝑡 , 𝑎𝑛𝑑 𝑧 = 𝑧 𝑡 . Then from the chain rule

𝑑𝑤

𝑑𝑡=

𝜕𝑓

𝜕𝑥

𝑑𝑥

𝑑𝑡+

𝜕𝑓

𝜕𝑦

𝑑𝑦

𝑑𝑡+

𝜕𝑓

𝜕𝑧

𝑑𝑧

𝑑𝑡

= 𝜕𝑓

𝜕𝑥𝒊 +

𝜕𝑓

𝜕𝑦𝒋 +

𝜕𝑓

𝜕𝑧𝒌 .

𝑑𝑥

𝑑𝑡𝒊 +

𝑑𝑦

𝑑𝑡𝒋 +

𝑑𝑧

𝑑𝑡𝒌 .

The first vector, denoted by grad𝑓 =𝜕𝑓

𝜕𝑥𝒊 +

𝜕𝑓

𝜕𝑦𝒋 +

𝜕𝑓

𝜕𝑧𝒌 is called the

gradient of the scalar function 𝑓 expressed in terms of Cartesian

coordinates, the second vector 𝑑𝒓

𝑑𝑡=

𝑑𝑥

𝑑𝑡𝒊 +

𝑑𝑦

𝑑𝑡𝒋 +

𝑑𝑧

𝑑𝑡𝒌 is seen to be a

vector that is tangent to the space curve Γ,consequently 𝑑𝑤

𝑑𝑡 is the scalar

product of grand 𝑓 and 𝑑𝒓

𝑑𝑡 at the point 𝑥 = 𝑥(𝑡), 𝑦 = 𝑦 𝑡 ,and 𝑧 = 𝑧 𝑡

for any given value of 𝑡.

Another notation for grad𝑓 is ∇𝑓 =𝜕𝑓

𝜕𝑥 𝒊 +

𝜕𝑓

𝜕𝑦𝒋 +

𝜕𝑓

𝜕𝑧 𝒌, where the symbol

∇𝑓 is either read “del f” or “grad f”. In this notation, the vector operator

∇= 𝒊𝜕

𝜕𝑥+ 𝒋

𝜕

𝜕𝑦+ 𝒌

𝜕

𝜕𝑧 is the gradient operator expressed in terms of

Cartesian coordinates, and if 𝜙 is a suitably differentiable scalar function

of 𝑥, 𝑦, and 𝑧.

∴ ∇𝜙 =𝜕𝜙

𝜕𝑥𝒊 +

𝜕𝜙

𝜕𝑦𝒋 +

𝜕𝜙

𝜕𝑧𝒌.

Geometrical representation of the gradient:

Let 𝜙 𝑝 = 𝜙 𝑥, 𝑦, 𝑧 be a differentiable scalar field. Let 𝜙 𝑥, 𝑦, 𝑧 = 𝑐

be a level surface and 𝑃0 𝑥0, 𝑦0 , 𝑧0 be a point on it. There are infinite

numbers of smooth curves on the surface passing through the point 𝑃0.

Each of these curves has a tangent at 𝑃0 the totality of all these tangent

lines from a tangent plane to the surface at the point 𝑃0. A vector normal

to this plane at 𝑃0 is called the normal vector to the surface at this point.

Consider now a smooth curve 𝐶 on the surface passing through a point 𝑃

on the surface. Let 𝑥 = 𝑥 𝑡 . 𝑦 = 𝑦 𝑡 , 𝑧 = 𝑧 𝑡 be the parametric

representation of the curve 𝐶. Any point 𝑃 on 𝐶 has the position vector

𝑟 = 𝑥 𝑡 𝑖 + 𝑦 𝑡 𝑗 + 𝑧 𝑡 𝑘 . Since the curve lies on the surface , we have

𝑓(𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 = 𝑘

Then 𝑑

𝑑𝑡𝑓(𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 = 0

By chain rule, we have 𝜕𝑓

𝜕𝑥

𝑑𝑥

𝑑𝑡+

𝜕𝑓

𝜕𝑦

𝑑𝑦

𝑑𝑡+

𝜕𝑓

𝜕𝑧

𝑑𝑧

𝑑𝑡= 0

Or 𝑖 𝜕𝑓

𝜕𝑥+ 𝑗

𝜕𝑓

𝜕𝑦+ 𝑘

𝜕𝑓

𝜕𝑧 . 𝑖

𝑑𝑥

𝑑𝑡+ 𝑗

𝑑𝑦

𝑑𝑡+ 𝑘

𝑑𝑧

𝑑𝑡 = 0

Or ∇𝑓. 𝑟′ 𝑡 = 0

Let ∇𝑓 𝑃 ≠ 0 and 𝑟′ 𝑡 ≠ 0.Now, 𝑟′ 𝑡 is a tangent vector to 𝐶 at the

point 𝑃 and lies in the tangent plane to the surface at 𝑃.Hence ∇𝑓 𝑃 is

orthogonal to every tangent vector at𝑃. Therefore, ∇𝑓 𝑃 is the vector

normal to the surface 𝑓 𝑥, 𝑦, 𝑧 = 𝑘 at the point 𝑃.

The unit normal vector is 𝒏 =grad 𝜑

grad 𝜑 .

Example: Find a unit normal vector to the surface 𝑥𝑦2 − 2𝑥𝑦𝑧 = 3 at the

point (1, 4, 3).

Solution: Let 𝜙 𝑥, 𝑦, 𝑧 = 𝑥𝑦2 − 2𝑥𝑦𝑧 = 3

⇒𝜕𝜙

𝜕𝑥= 𝑦2 − 2𝑦𝑧,

𝜕𝜙

𝜕𝑦= 2𝑥𝑦 − 2𝑥𝑧 and

𝜕𝜙

𝜕𝑧= −2𝑥𝑦.

∴ ∇𝜙 =𝜕∅

𝜕𝑥𝒊 +

𝜕∅

𝜕𝑦𝒋 +

𝜕∅

𝜕𝑧𝒌

= 𝑦2 − 2𝑦𝑧 𝒊 + 2𝑥𝑦 − 2𝑥𝑧 𝒋 − 2𝑥𝑦𝒌

∇𝜙 1,4,3 = −8𝒊 + 2𝒋 − 8𝒌.

The unit normal vector at (1,4,3) is

𝒏 =−8𝒊+2𝒋−8𝒌

64+4+64=

2 −4𝒊+𝒋−4𝒌

132.

Note: The angle between two surfaces 𝜙1 𝑥, 𝑦, 𝑧 and 𝜙2 𝑥, 𝑦, 𝑧 at a

point is the angle between their norals at that point.

𝑋

𝑍

𝒓′(𝑡)

𝑜

∇𝜙(𝑃0)

Γ

𝑌

i.e. cos 𝜃 = 𝒏𝟏 .𝒏𝟐

𝒏𝟏 . 𝒏𝟐 . Where 𝒏𝟏 =

∇𝜙1

∇𝜙1 , 𝒏𝟐 =

∇𝜙2

∇𝜙2 .

Properties:

Let 𝑓 and 𝑔 be any two differentiable scalar fields. Then

∇ f ± g = ∇f ± ∇g.

∇ 𝑐1𝑓 + 𝑐2𝑔 = 𝑐1∇𝑓 + 𝑐2∇𝑔, 𝑐1 , 𝑐2 are arbitrary constants.

∇ fg = f∇g + g∇f.

∇ f

g =

g∇f−f∇g

g2 , g ≠ 0.

Directional derivative:

Consider a scalar function 𝜙 𝑥, 𝑦, 𝑧 with continuous first order partial

derivatives with respect to x, y, and z that is defined in some region of

space. The partial derivatives 𝜕∅

𝜕𝑥,𝜕∅

𝜕𝑦 and

𝜕∅

𝜕𝑧 can be interpreted as the slope

(or rate of change) of 𝜙 𝑥, 𝑦, 𝑧 along 𝒊, 𝒋 and 𝒌 directions, respectively.

We can also evaluate derivatives along intermediate between 𝒊, 𝒋 and 𝒌.

The result is a directional derivative. The directional derivative of 𝜙 at a

point 𝑝(𝑥, 𝑦, 𝑧) in the direction of a unit vector 𝒖 is denoted by 𝐷𝒖 𝜙 .

The directional derivative is the slope (or rate of change) of 𝜙 𝑥, 𝑦, 𝑧 in

the direction of 𝒖.

𝐷𝒖 𝜙 = ∇𝜙. 𝒖.

Example: Find the directional derivative of 𝜙(𝑥, 𝑦, 𝑧) = 𝑥2 + 3𝑦2 + 2𝑦2

in the direction of the vector 2𝒊 − 𝒋 − 2𝒌 and determine its value at the

point 1, −3,2 .

Solution: ∇𝜙 = 2𝑥𝒊 + 6𝑦𝒋 + 4𝑧𝒌 and the unit vector is

𝒖 =2𝒊−𝒋−2𝒌

4+1+4=

2𝒊−𝒋−2𝒌

9=

1

3(2𝒊 − 𝒋 − 2𝒌).

Directional derivative of 𝜙 in the direction 𝒖

𝐷𝒖 𝜙 = ∇∅. 𝒖

= 2𝑥𝒊 + 6𝑦𝒋 + 4𝑧𝒌 .1

3 2𝒊 − 𝒋 − 2𝒌

=4

3𝑥 − 2𝑦 −

8

3𝑧.

Directional derivative 𝐷𝒖 𝜙 at the point 1, −3,2 is

𝐷𝒖 𝜙 1,−3,2 =4

3+ 6 −

16

3= 2.

Properties:

The most rapid increase of a differentiable function 𝜙 𝑥, 𝑦, 𝑧 at a

point 𝑝 in space occurs in the direction of the vector 𝒖𝑝 =

grad𝜙 𝑝 . The directional derivative at 𝑝 is

𝐷𝒖𝜙 𝑝 = grad𝜙 𝑝 = 𝜕𝜙

𝜕𝑥

𝑝

2+

𝜕𝜙

𝜕𝑦

𝑝

2+

𝜕𝜙

𝜕𝑧

𝑝

2

1

2

.

The most rapid decrease of a differentiable function 𝜙 𝑥, 𝑦, 𝑧 at a

point 𝑝 in space occurs when the vector 𝒖𝑝 just defined in 𝑖 and

grad𝜙 are oppositely directed , so that 𝒖𝑝 = −grad𝜙 𝑝 the

directional derivative at 𝑝 is

𝐷𝒖𝜙 𝑝 = − grad𝜙 𝑝 = − 𝜕𝜙

𝜕𝑥

𝑝

2+

𝜕𝜙

𝜕𝑦

𝑝

2+

𝜕𝜙

𝜕𝑧

𝑝

2

1

2

.

There is a zero local rate of change of a differentiable function

𝜙 𝑥, 𝑦, 𝑧 at a point 𝑝 in space in the direction of any vector 𝒖𝑝 that

is orthogonal to grad𝜙 at 𝑝 , so that 𝒖𝑝 .grad 𝜙 𝑝 = 0.