Notes notes vector calculus made at home (wecompress.com)

VECTOR CALCULUS FROM PAGE 16 – PARTIAL DERIVATIVE
Vector Calculus Lecture Notes 2015 GREAT PROFESSOR Andrea
Moiola University of Reading 23 Sep 2015 11

{
Position Vector r equals xi + yj + zk
Fields are functions of position described by position vector so their domain is
Euclidean space of subset of Euclidean Space R3
}

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SAME AS
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SAME AS
))
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SAME AS
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NOTES NOTES VECTOR CALCULUS MADE AT HOME
NOTES NOTES VECTOR CALCULUS MADE AT HOME
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VECTOR CALCULUS ANDREA MOIOLA
PAGE 2
We use the hat symbol (ˆ) to denote unit vectors, i.e. vectors of length 1.
i j k with hat symbol denote vectors
i j k are three fixed vectors that constitute the canonical basis of Euclidean
Space viz R3
Vector u viz

Magnitude of Vector and Direction of Vector
Length and direction uniquely identify a vector
Every vector satisfies ~u = |~u|ˆu. Therefore length and direction uniquely
identify a vector.
The vector of length 0 (i.e. ~0 := 0ˆı + 0ˆj + 0ˆk) does not have a specified
direction
POSITION VECTOR
Vectors defined as geometric entities fully described by magnitude and direction are
sometimes called “Euclidean vectors” or “geometric vectors”
Page 3

⋆ Remark 1.5. The addition, the scalar multiplication and the scalar product are defined
for Euclidean spaces of any dimension, while the vector product (thus also the triple
product) is defined only in three dimensions.
The addition, the scalar multiplication and the scalar product are defined for Euclidean
spaces of any dimension
Vector product thus also the triple product is defined only in three dimensions
)
(
IMPORTANT RESULT
PAGE 6
)
(
Gradient is Vector field viz partial derivative of Scalar Field

Directional Derivative is product of Unit Vector and Gradient
Normal derivative viz unit vector is orthogonal to surface then derivative of
scalar field
)

(
VERY IMPORTANT PROPERTIES OF GRADIENT
)
THE REAL BOOK Mathematics
PAGE 183 OF 2321 – THE REAL BOOK

Acceleration due to gravity or velocity of fluid examples of vector
field
Vector product used to compute angular momentum of moving
object Torque of force Lorentz force acting on charge moving in
magnetic field
Vector product used to compute angular momentum of moving
object, Torque of force, Lorentz force acting on charge moving in
magnetic field
Partial Derivatives of Scalar Fields

(
Partial Derivatives of Vector Fields
)
(
Directional Derivative : The component of Delta f in any direction is rate of
change of f in that direction
Page 185 & 186
DIRECTIONAL DERIVATIVE

Gradient is orthogonal to tangent plane and hence to the surface

(
THE REAL BOOK
PAGE 186 OF 2321
Gradient is Orthogonal to Tangent Plane and hence to Surface
)
(
Page 186
Tangent Plane to Surface at point
)

(
Page 187
Gradient of function viz Delta of Function is in direction of maximum
directional derivative
Magnitude of Gradient is value of directional derivative in that
direction
Directional Derivative defined viz
Gradient of function is Zero the Gradient is in direction of maximum
directional derivative
Magnitude of Gradient viz Mod of Gradient is value of directional
derivative in that direction
Gradient points in uphill direction and magnitude of gradient is uphill
slope
Value of Directional Derivative is Mod of Gradient
Directional Derivative θ angle between Gradient and Terrain
)

(
Curl of vector field F
Curl is operated on Vector Field
CURL OF VECTOR

((
Irrotational Vector viz Curl of Vector equals zero
Solenoidal Vector viz divergence free or incompressible vector
Conservative Vector Field
Scalar potential of Vector Field
Vector Potential of Vector Field
Irrotational Vector viz Curl of Vector equals zero

))
(
Gradient vis a vis Divergence
Divergence vis a vis Gradient
Gradient is a vector and Divergence is a scalar
More specifically (and perhaps helpfully), the gradient vector points in the
direction of the fastest (local) increase in the value of the (scalar) function.
)

((((
GREAT PROFESSOR ANDREA MOIOLA UNIVERSITY OF READING –
VECTOR CALCULUS
GREAT PROFESSOR ANDREA MOIOLA UNIVERSITY OF READING –
VECTOR CALCULUS
(
FIELDS FIELDS
FIELDS FIELDS SCALAR FIELDS VECTORS FIELDS
Page 7 8 9
Vector field is a function of position that assigns a vector to each
point in space
r is point
r is point
Fields are functions of position described by the position vector r = xi + yj
+ zk
Fields domain is Euclidean Space R3
or subset of it
Fields depending on the kind of output they are called either scalar fields
or vector fields
Vector functions are vector valued functions of a real variable.
F capital denotes Vector field
point in space
Scalar Field is function where domain D is an open subset of
Euclidean Space

Scalar functions in particular real functions of a real variable viz rules that
associate to every number t ∈ R a second number f(t) ∈ R.
Scalar functions in particular real functions of a real variable viz
rules that associate to every number Real Number to second Real
number
Vector functions whose domains or codomains or both are three
dimensional Euclidean space viz Real Number cubed as opposed to
real line R
Three different extensions of concept to Vector case viz consider functions
whose domains or codomains or both are three dimensional Euclidean
space as opposed to the real line R.
.
Remark 1.27 we summarise the mapping properties of all these objects.
Scalar fields, vector fields and
vector functions are described in [1] in Sections 12.1, 15.1 and 11.1
respectively.
Euclidean Space
Scalar field is a function where domain d is an open subset of
Euclidean Space
Scalar field value viz Value of function at the point r may be written
as f(r) or f(x y z)
r is point
Scalar field can equivalently be interpreted as a function of one
vector variable or as a function of three scalar variables

Scalar field may also be called multivariate functions or functions of
several variables
Examples of Scalar Fields are
1.2.1 Scalar fields
A scalar field is a function f : D → R, where the domain D is an open
subset of R3. The value of f at the
point ~r may be written as f(~r) or f(x, y, z). (This is because a scalar field
can equivalently be interpreted as a function of one vector variable or as a
function of three scalar variables.) Scalar field may also be called
“multivariate functions” or “functions of several variables” (as in the first-
year calculus modulus, see handout 5). Some examples of scalar fields
are
f(~r) = x2 − y2, g(~r) = xyez, h(~r) = |~r|4.
Two dimensional may also be thought as Three Dimensional fields
that do not depend on Third Variable
Two different scalar fields may have the same level surfaces
associated with different field
Vector functions are vector valued functions of a Real Variable
)
CURVE DEFINITION LOOP DEFINITION
If variable of which Vector function is comprised is continous viz its
three components are continous real functions then we call it Curve

If the interval of Vector function is closed and bounded then vector
valued function of real variable is Loop
Injective viz of the nature of or relating to an injection or one-to-one
mapping.
Curve indicates a function of real variable whose image its path is a
subset of R3
and not the image itself
Indeed, different curves may define the same path
Curve” indicates a function ~a, whose image (its path) is a subset of R3,
and not the image itself. Indeed, different curves may define the same
path
Function space is an infinite dimensional vector space whose
elements are functions
F capitalised F is symbol for vector field
(
SCALAR PRODUCT
)

(
VECTOR PRODUCT
PAGE 4
Vector product is distributive with respect to sum but is not associative
)

((
VECTOR PRODUCT OF TWO IDENTICAL VECTORS IS ZERO
VECTOR PRODUCT OF TWO EQUAL VECTORS IS ZERO
CROSS PRODUCT OF TWO IDENTICAL VECTORS IS ZERO
CROSS PRODUCT OF TWO EQUAL VECTORS IS ZERO
https://www.quora.com/If-two-vectors-are-equal-then-what-will-be-
their-cross-product
))

(
BEAUTIFUL MATHEMATICS BEAUTIFUL EXAMPLE
PAGE 5 & 83
)
(
PAGE 5 & 83
Vector product j x j equals zero viz equals zeroz
j component is “0 1 0” viz i j k equals “0 1 0”
Product of j x j equals “0 1 0” x “0 1 0” equals zeroz
)
(

PAGE 5 & 83
VERY IMPORTANT RESULT IN VECTOR CALCULUS
Vector u x (v x w) = v(u.w) – w(u.v)
)

(
Magnitude of Vector Product
)
(
VERY IMPORTANT EXAMPLE
Sum of Perpendicular and Parallel Vectors
Page 4
Page 83
)

(
Vector product is not associative
)
Jacobi Identity
Binet Cauchy Identity
Lagrange Identity
GREAT PROFESSOR ON VECTOR CALCULUS
))

(
REFER BOOK
Page 7 of 112
COMPLEMENT OF OPEN SET IS CLOSED AND VICE VERSA
COMPLEMENT OF CLOSED SET IS OPEN AND VICE VERSA
)

(
Angle θ between Tangent Planes to Surface:
Angle Theta between Tangent Planes to Surface
)
(
Vector Space
Vector space viz space consisting of vectors together with the
associative and commutative operation of addition of vectors and
the associative and distributive operation of multiplication of vectors
by scalars
EUCLIDEAN SPACE DEFINED
VECTORS are in Euclidean space R3
If the considered vector space is real, finite-dimensional and is provided
with an inner product, then it is an Euclidean space (i.e., Rn for some
natural number n).
If a basis is fixed, then elements of Rn can be represented as n-tuples of
real numbers (i.e., ordered
sets of n real numbers).
)

Scalar multiplication and the scalar product are defined for Euclidean
spaces of any dimension, while the vector product (thus also the triple
product) is defined only in three dimensions.
Scalar Product

Orthogonal or Perpendicular Vectors viz Scalar Product is zero
Parallel vectors
Vector function take real number as input and return a Vector
Vector fields might be thought as combinations of three scalar fields
viz the components
Vector fields might be thought as combinations of three scalar fields
viz three functions whose domain D are an open subset of Euclidean
Space
Euclidean Space
Euclidean Space
Vector functions as combinations of three real functions

(
Differentiable Scalar Field
Differentiable Vector Field
)

(
ORTHOGONAL FUNCTIONS INNER PRODUCT IS ZERO
TWO FUNCTIONS ARE ORTHOGONAL IF INNER PRODUCT OF
THESE TWO FUNCTIONS IS ZERO

(
Scalar Fields and Gradients comparision
Similarity between Function Derivative and Scalar Fields and their
Gradients
Relations between functions and derivative carry over to Scalar
Fields and their Gradients
Page 14
)

(
JACOBIAN MATRIX
F capital is Vector Field
Vector field has nine partial derivatives three of each component viz i
j k
point in space
Jacobian Matrix is Matrix of nine partial derivatives of VECTOR
FIELD
Scalar field has three first order partial derivatives that can be
collected in a vector field
Vector field has nine partial derivatives three for each component
which is scalar field to represent them compactly we collect them in
a matrix called Jacobian Matrix
Vector field has nine partial derivatives three for each component
which is scalar field

Second order derivative of scalar field in Hessian viz Jacobian gradient of
scalar field
Whereas Jacobian of Vector field is first order derivative

)
((
VERY IMPORTANT EXAMPLE
Product of Orthogonal functions is Zero
Inner Product of Orthogonal functions is Zero
Orthogonal Functions need not be with one gradient of other
Page 15 and 85

))
((
LAPLACIAN viz smooth Scalar field obtained as sum of pure second
partial derivatives of function f
Scalar field whose Laplacian vanishes everywhere is called harmonic
function
Scalar field whose Laplacian vanishes viz second order derivative is
zero everywhere is called harmonic function since there is no
dissonance
Scalar field whose Laplacian vanishes everywhere is called harmonic
function viz No Dissonance
HARMONIC : an overtone accompanying a fundamental tone at a fixed interval,
produced by vibration of a string, column of air, etc. in an exact fraction of its
length

Hessian Matrix is second derivative of Scalar Field f
Laplacian equals trace of Hessian viz sum of diagonal terms viz
Laplacian is second order derivative of Gradient viz above
Hessian Matrix is second derivative of Scalar Field f
Hessian Matrix is Jacobian Matrix of Gradient

Vector Laplacian is Laplace operator applied componentwise to
Vector fields
))

((
DIVERGENCE DIVERGENCE DIVERGENT DIVERGENT
Divergent is always applied on vector field and resultant is scalar
Solenoidal Vector viz Divergence of vector equals zero
))))

’))
((
CURL CURL
Curl maps vector fields to vector fields
CURL OPERATOR
))

((
Scalar fields take as input vector and return a real number
Vector fields take as input vector and return a vector
Vector functions take as input real number and return a vector
))

((
PAGE 21
Divergence is Trace of Jacobian Matrix
Jacobian Matrix of Gradient is Hessian
Trace of Hessian is Laplacian
Divergence is Trace of Jacobian Matrix
Jacobian Matrix of Gradient is Hessian
Trace of Hessian is Laplacian
))

(
VECTOR PRODUCT OR CROSS PRODUCTS
https://betterexplained.com/articles/cross-product/
Integrals are "multiplication, taking changes into account" and the dot product is
"multiplication, taking direction into account".
(
MAGNITUDE OF VECTOR PRODUCT VIZ CROSS PRODUCT:
VECTOR PRODUCT CROSS PRODUCT
)

((
VECTOR PRODUCT OR CROSS PRODUCT
Magnitude of Vector Product or Cross Product
Page 4 of 112
))

((
REFERENCE Vector Calculus Revision of Basic Vectors
Page 1 of 17
Scalar Product or Dot Product
Vector Product or Cross Product
Scalar Triple Product
Vector Triple Product

))
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Angle Between Two Surfaces
Angle Between Two Vectors

Projection of Vector
))
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VECTOR INTEGRATION
Maths 1 Vector Calculus by Y Prabhaker Reddy
Page 11 of 15
r co-ordinate in 3D viz Euclidean Space
F is field vector

))
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VECTOR SURFACE INTEGRAL
Page 12 of 15

((
VECTOR SURFACE INTEGRAL AS PROJECTION OF
SURFACE ON XY YZ XZ PLANE
Page 12 of 15
))
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VOLUME INTEGRAL OF VECTOR FIELD
Page 13 of 15
))

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VOLUME INTEGRAL OF SCALAR FIELD
Page 13 of 15
))

((
VECTOR INTEGRATION
Page 13 14 15 of 15
Green’s Theorem for Vector Integration

Gauss Divergence Theorem for Vector Integration
Stokes Theorem for Vector Integration
))

((
Advection Operator applied to Vector Field
Page 23 of 112
))
((
Derivative of Vector Function is Vector Function
Page 26 of 112
))

((
Curve is SMOOTH if its three components are smooth functions
Page 26 of 112
))
((
DERIVATIVE OF COMPOSITE FIELD SCALAR FIELD AND VECTOR FIELD
USING CHAIN RULE

Partial Derivatives of composition of Scalar Field with Vector Field
Scalar field as composite of scalar and vector field
(
Below Derivatives are special cases of a more general result for functions
between Euclidean spaces of Arbitrary Dimensions

((
VECTOR CALCULUS BY ANDREA MOIOLA
PAGE 27 OF 112
Scalar field f evaluated on smooth curve a viz f(a)
Scalar field f equals xyez
and curve a equals ti + t3
j
Compute total derivate of f(a)
Scalar field f evaluated on smooth curve a viz f(a)

Since the Scalar field f evaluated on smooth curve a viz f(a) viz 3D filed
evaluated on 2D smooth curve the Gradient of f has i component has yez
which
corresponds to yth componet of curve viz t3
viz yez
= t3
Gradient of f has i component yez
which corresponds to yth componet of curve
viz t3
viz yez
= t3
There is a strong emphasis on f((a)t) viz Scalar field f evaluated on smooth
curve a therefore gradient of “f” in terms of “a” will have x and y co-ordinates
in terms of curve “a”
Gradient of f has j component xez
which corresponds to xth componet of curve
viz t viz xez
= t
Since 3D function f is expressed in terms of 2D curve viz gradient of function with x
coordinate corresponds to x coordinate of “a” and gradient of function with y
coordinate corresponds to y coordinate of “a”

((
Derivative of Vector field g(r) constrained to 2d SURFACE
Vector field g(r) and position vector “r”
PAGE 28 OF 112
Vector field g(r) and position vector “r”
Value of field g depends upon Sh viz position vector
))
((
VERY VERY IMPORTANT RESULTS
Partial Derivatives and Total Derivatives of Composite Functions
Given Curve Scalar field and Vector field F = xi + zj –yk compute
Partial Derivatives and Total Derivatives of Composite Functions

Given Curve a(t) Scalar field “g” and Vector field F = xi + zj –yk
compute partial derivatives of gF and total derivatives of ga and Gfa
by calculating compositions and then deriving them and by using
Vector formulas
PAGE 29 OF 112
CONSIDER g scalar function composite of Vector Field F and Curve “a”
Curve “a” pertains to x y co-ordinate therefore F vector field evaluated on
Curve “a” has sint j as y component corresponds to y coordinate on F
vector field viz yk
Therefore

))
END END GREAT PROFESSOR ANDREA MOIOLA UNIVERSITY OF
READING – VECTOR CALCULUS
))))
))))))))

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