Derivative of Vector valued Functions
Let 𝑟 𝑡 be the position vector of a point P[𝑥 𝑡 , 𝑦 𝑡 , 𝑧 𝑡 ] in
space, where t is scalar quantity time.
Then 𝑟 𝑡 in the component form is
𝑟 𝑡 =𝑥 𝑡 i + 𝑦 𝑡 j + 𝑧 𝑡 k
𝑑𝑟
Velocity: Derivative of 𝑟 𝑡 with respect to time t , that is,
𝑑𝑡
𝑑𝑟
is called the velocity and given by 𝑣 = .
𝑑𝑡
Acceleration: Derivative of 𝑣 𝑡 with respect to time t , that
𝑑𝑣
is, is called the acceleration and given by
𝑑𝑡
𝑑𝑣 𝑑2𝑟
𝐴= or 𝐴 = .
𝑑𝑡 𝑑𝑡 2
1. Component of a vector( 𝑣 ) or
acceleration(𝐴) along a given vector along the
tangent or along the direction 𝐷 is the
resolved part of 𝑣 or 𝐴, given by 𝑣. 𝑛 or 𝐴. 𝑛
𝐷
where 𝑛 =
𝐷
2. Component of a vector( 𝑣 ) or acceleration(𝐴)
along the Normal or along perpendicular to the
direction 𝐷 is given by
= 𝑣 − 𝑣. 𝑛 𝑛 𝑜𝑟 𝐴 − (𝐴. 𝑛)𝑛
𝐷
where 𝑛 =
𝐷
Unit Tangent vector and unit normal vector
Let A be a fixed point on the curve and let the length of the
arc AP be equal s,
where P= (x, y, z) and we have 𝑟 = 𝑥 𝑡 𝑖 + 𝑦 𝑡 𝑗 + 𝑧(𝑡)𝑘
𝑑𝑟
The tangent vector is 𝑇 =
𝑑𝑡
𝑑𝑟 𝑑𝑟 𝑑𝑟
Unit tangent vector 𝑇 = = / =𝑇/ 𝑇
𝑑𝑠 𝑑𝑡 𝑑𝑡
𝑑𝑇
The normal vector is 𝑁 =
𝑑𝑠
𝑑𝑇 𝑑𝑟
The unit normal vector 𝑁 = / which is also called as
𝑑𝑡 𝑑𝑡
the principal normal vector.
QUESTION
1-A particle moves along a curve whose parametric equations
are 𝑥 = 𝑒 −𝑡 , 𝑦 = 2 cos 3𝑡, 𝑧 = 2𝑠𝑖𝑛3𝑡 where t is the time.
Find the velocity and the acceleration at any time t and also
their magnitudes at t=0.
2-A particle moves along the curve
𝑥 = 2𝑡 2 , 𝑦 = 𝑡 2 − 4𝑡, 𝑧 = 3𝑡 − 5 where t is the time. Find the
components of its velocity and acceleration in the direction of
𝑖 − 2𝑗 + 2𝑘 at t=1.
3-Determine the unit normal vector to the surface 𝑥 2 𝑦 − 2𝑥𝑧 + 2𝑦 2 𝑧 4 =
10 at (2, 1, -1).
Scalar Point Functions
If to every point P[𝑥, 𝑦, 𝑧] of a region R in space, there
corresponds a scalar function ∅(𝑥, 𝑦, 𝑧), then ∅ is called scalar
point function. Ex : ∅ = 𝑥 2 𝑦z or xyz
Vector Point Functions
If to every point P[𝑥, 𝑦, 𝑧] of a region R in space, there
corresponds a vector 𝑅(𝑥, 𝑦, 𝑧), then 𝑅 is called vector point
function. Ex : 𝑅 = 𝑥 2 𝑦𝑖 + 𝑥𝑦 𝑗 +xz k or xy i + yz j +xyz k
Del Operator: The vector differential operator 𝛻 is defined as
𝜕 𝜕 𝜕
𝛻= 𝑖+ 𝑗+ 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
Gradient of Scalar Field
If ∅ 𝑥, 𝑦, 𝑧 is a continuously differentiable scalar function,
then gradient of ∅ or written as grad ∅ , is defined as
𝜕∅ 𝜕∅ 𝜕∅
grad ∅=𝛻∅ = 𝑖 + 𝑗 + 𝑘
𝜕𝑥 𝜕𝑥 𝜕𝑥
It is a vector point function.
Geometrical Meaning
If ∅ 𝑥, 𝑦, 𝑧 is a scalar point function, then grad ∅ is vector
normal to the surface ∅ 𝑥, 𝑦, 𝑧 = 𝑐, 𝑐 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
Angel between Two Surfaces: If ∅1 𝑥, 𝑦, 𝑧 = 𝑐1 and
∅2 𝑥, 𝑦, 𝑧 = 𝑐2 are two surfaces, then angle between them is
1 2
cos
1 2
Directional Derivative
If ∅ 𝑥, 𝑦, 𝑧 is a scalar point function and 𝐷 is the given
direction, then
𝐷
Directional derivative of ∅ along 𝑛 =𝛻∅. 𝑛, where 𝑛 =
𝐷
Maximum Directional Derivative
The directional derivative is maximum along 𝛻∅ and its
maximum value is equal to 𝛻∅ 𝑜𝑟 𝑔𝑟𝑎𝑑 ∅ .
QUESTION
Find the directional derivative of the following 𝜑 = 𝑥 2 𝑦𝑧 +
4𝑥𝑧 2 at (1, -2, -1) along 2𝑖-𝑗-2𝑘.
QUESTION
In which direction the directional derivative of 𝑥 2 𝑦𝑧 3 is
maximum at (2, 1, -1) and examine the magnitude of
this maximum.
Divergence
Divergence of a vector point function 𝑅(𝑥, 𝑦, 𝑧), is written
𝜕 𝜕 𝜕
div𝑅= 𝛻. 𝑅= 𝑖 + 𝑗 + 𝑘 . 𝑅1 𝑖 + 𝑅2 𝑗 + 𝑅3 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝑅1 𝜕𝑅2 𝜕𝑅3
div𝑅 = + +
𝜕𝑥 𝜕𝑦 𝜕𝑧
It is a scalar quantity.
Solenoidal Vector: A vector is said to be solenoidal vector if
it’s divergence is zero, that is div 𝑅 = 0.
CURL
Curl of a vector point function 𝑅(𝑥, 𝑦, 𝑧), is written
𝜕 𝜕 𝜕
Curl 𝑅= 𝛻 × 𝑅= 𝑖 + 𝑗 + 𝑘 × 𝑅1 𝑖 + 𝑅2 𝑗 + 𝑅3 𝑘
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝑖 𝑗 𝑘
𝜕 𝜕 𝜕
Curl 𝑅 = 𝜕𝑥 𝜕𝑦 𝜕𝑧
𝑅1 𝑅2 𝑅3
It is a vector quantity.
Irrotational Vector: A vector is said to be irrotational vector
if it’s curl is zero, that is curl 𝑅 = 0.
