PPT: Elements of Vector Calculus | GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE) PDF Download

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 Page 1


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
Page 2


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Page 3


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Page 4


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Gradient
 Component of gradient is the partial derivative
in the direction of that component
 Fourier´s Law of Heat Conduction
s
s
¶
¶
= Ñ
f
f . e
n . T k
n
T
k q Ñ - =
¶
¶
- =
&
f Ñ
s
s e ,
Page 5


In 1-D
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
In 3-D
Differential Calculus w.r.t. Space
Definitions of div, grad and curl
t 0
?S
1
div lim . dS
t
d
d
®
æö
º
ç÷
èø
ò
D Dn
Ñ
t 0
?S
1
curl lim dS
t
d
d
®
æö
º-´
ç÷
èø
ò
D Dn
Ñ
Elemental volume dt
with surface DS
n
dS
D=D(r), f= f(r)
[ ]
x0
1
lim ( ) ()
df
fx x fx
dxx
D®
æö
= +D-
ç÷
D
èø
fndS
(large)
Gradient
t 0
?S
1
grad lim dS
t
d
ff
d
®
æö
º
ç÷
èø
ò
n
Ñ
dS
n
Elemental volume dt
with surface DS
fndS
(small)
fndS
(medium)
fndS
(medium)
= magnitude and direction of the slope in the scalar field at a point
Resulting fndS
Review
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Gradient
Magnitude and direction
of the slope in the scalar
field at a point
Gradient
 Component of gradient is the partial derivative
in the direction of that component
 Fourier´s Law of Heat Conduction
s
s
¶
¶
= Ñ
f
f . e
n . T k
n
T
k q Ñ - =
¶
¶
- =
&
f Ñ
s
s e ,
z y x
dxdydz
z
dxdydz
y
dxdydz
x ¶
¶
+
¶
¶
+
¶
¶
=
÷
÷
ø
ö
ç
ç
è
æ
¶
¶
+
¶
¶
+
¶
¶
=
º
®
®
ò
f f f f f f
d
f
d
f
d
d
k j i k j i
n
t
1
Lim
dS
t
1
Lim grad
0 t
?S
0 t
Face 2
Differential form of the Gradient
ò
®
º
?S
0 t
dS
t
1
Lim grad n f
d
f
d
Cartesian system
dy
dx
dz
j
i
k
P
Evaluate integral by expanding the variation in
f about a point P at the center of an elemental
Cartesian volume. Consider the two x faces:
f = f(x,y,z)
Face 1
dydz
dx
x
Face
) (
2
dS
1
i n -
÷
ø
ö
ç
è
æ
¶
¶
- »
ò
f
f f
dydz
dx
x
Face
) (
2
dS
2
i n +
÷
ø
ö
ç
è
æ
¶
¶
+ »
ò
f
f f
adding these gives
dxdydz
x ¶
¶ f
i
Proceeding in the same way for y and z
dxdydz
y ¶
¶ f
j dxdydz
z ¶
¶ f
k and we get , so
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FAQs on PPT: Elements of Vector Calculus - GATE Notes & Videos for Electrical Engineering - Electrical Engineering (EE)

1. What is vector calculus and why is it important in mathematics?
Ans. Vector calculus is a branch of mathematics that deals with differentiation and integration of vector fields. It is used to study and analyze physical phenomena that involve quantities with both magnitude and direction, such as velocity, force, and electromagnetic fields. Vector calculus is essential in various fields of science and engineering, including physics, computer graphics, fluid dynamics, and electromagnetism.
2. What are the key elements of vector calculus?
Ans. The key elements of vector calculus include vector fields, line integrals, surface integrals, divergence, curl, gradient, and the fundamental theorems of calculus. Vector fields represent the distribution of vectors in space, while line integrals and surface integrals quantify the flow of vectors along curves and surfaces, respectively. Divergence measures the tendency of a vector field to originate or converge at a point, curl represents the rotation of a vector field, and gradient measures the rate of change of a scalar function.
3. How is vector calculus applied in physics?
Ans. Vector calculus is extensively applied in physics to describe and analyze various physical phenomena. For example, in electromagnetism, vector calculus is used to model and understand electric and magnetic fields, electromagnetic waves, and Maxwell's equations. In fluid dynamics, vector calculus is employed to study fluid flow, turbulence, and conservation laws. Additionally, vector calculus plays a crucial role in mechanics, quantum mechanics, and general relativity.
4. What are the fundamental theorems of vector calculus?
Ans. The fundamental theorems of vector calculus include the divergence theorem and Stokes' theorem. The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field within the enclosed region. Stokes' theorem relates the circulation of a vector field around a closed curve to the surface integral of the curl of the field over any surface bounded by that curve. These theorems provide important relationships between line integrals, surface integrals, and volume integrals.
5. Can you provide an example of a real-life application of vector calculus?
Ans. One example of a real-life application of vector calculus is in fluid dynamics, particularly in the study of airflow around objects. By using vector calculus, engineers can analyze the velocity and pressure fields around an airplane wing, for instance, to optimize its aerodynamic performance. Vector calculus is also employed in computer graphics and animation, where it is used to model the movement and deformation of objects in three-dimensional space.
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