This Downward Pointing Triangle Means Grad Div and Curl in Vector Calculus (Nabla / Del) by Parth G
Summary
TLDRThis video introduces important differential operators—gradient, divergence, and curl—commonly used in physics and vector calculus. It explains how the 'nabla' operator is applied to scalar and vector fields, demonstrating how these operators measure changes in different directions. Through clear examples, such as flour distribution and fluid flow, the video delves into the concepts of gradient (change in a scalar field), divergence (outflow or inflow of a vector field), and curl (rotation in a vector field). The speaker also links these concepts to Maxwell's equations in electromagnetism.
Takeaways
- 🔺 Nabla (or del) is a vector used to represent differential operators like gradient, divergence, and curl.
- 🔄 Gradient measures how quickly a scalar field changes in space and results in a vector field.
- 📏 Partial derivatives (denoted by ∂) help isolate the change in a specific direction while keeping others constant.
- 🌍 Scalar fields represent quantities like altitude or temperature, and the gradient of these fields indicates the direction of the steepest ascent.
- 🌬️ Vector fields can represent physical phenomena like wind or electric fields, assigning a vector to each point in space.
- ⚡ Divergence measures how much a vector field spreads out from a point and can indicate sources or sinks in fields like electric fields.
- 💥 In electromagnetism, the divergence of an electric field relates to the presence of electric charges, and divergence of magnetic fields is always zero (no magnetic monopoles).
- 🔁 Curl measures the rotation or circulation of a vector field, such as the rotation of fluid flow, and results in another vector field.
- 📐 In physics, the curl operator is used in Maxwell’s equations, showing how changing magnetic fields induce electric fields.
- 🎓 Gradient, divergence, and curl are essential tools in vector calculus, with applications in physics like gravity, electromagnetism, and fluid dynamics.
Q & A
What are the gradient, divergence, and curl operators, and where are they commonly used?
-The gradient, divergence, and curl operators (often called grad, div, and curl) are differential operators used in vector calculus. They are widely used in physics, particularly in areas like fluid dynamics, electromagnetism, and gravitational fields. The gradient measures the rate of change of a scalar field, the divergence measures how much a vector field spreads out from a point, and the curl measures the rotation or circulation of a vector field.
What is the del (nabla) operator, and how does it relate to partial derivatives?
-The del (nabla) operator, denoted by a downward-pointing triangle, acts like a vector of partial derivatives in three dimensions. Its components are partial derivatives with respect to x, y, and z. It measures how quickly a quantity changes over small distances in different directions.
How does the gradient operator work, and what does it represent?
-The gradient operator, when applied to a scalar field, gives a vector field that represents the direction and rate of fastest change of the scalar field. For example, in a topographical map, the gradient would indicate the direction of the steepest incline at each point.
What is the physical interpretation of the divergence of a vector field?
-The divergence of a vector field gives a scalar value that represents how much the field is spreading out or converging at a point. For instance, in fluid flow, a positive divergence would indicate that fluid is flowing out from a point (a source), while a negative divergence indicates fluid flowing into a point (a sink).
How does the curl operator work, and what does it measure?
-The curl operator, when applied to a vector field, produces another vector field that represents the rotation or circulation of the original vector field. The direction of the resulting vector indicates the axis of rotation, while the magnitude represents the strength of the rotation.
Can you explain the physical significance of scalar and vector fields?
-A scalar field assigns a single value to each point in space (e.g., temperature distribution), while a vector field assigns a vector to each point (e.g., wind speed and direction). Scalar fields measure quantities like height or temperature, while vector fields measure directional phenomena like electric fields or fluid flow.
How is the gradient of a scalar field applied in physics, particularly with gravitational fields?
-In physics, the gradient of a scalar field, such as the gravitational potential, gives a vector field representing the gravitational force. The direction of the gradient indicates the direction in which the gravitational force acts, while its magnitude shows the strength of the force.
What role does the dot product play in calculating divergence?
-The dot product in the context of divergence applies the del operator to a vector field. It combines the partial derivatives from the del operator with the components of the vector field, resulting in a scalar field that measures how much the field is expanding or contracting at a point.
How are Maxwell's equations related to the divergence and curl operators?
-Maxwell's equations, which describe electric and magnetic fields, make extensive use of divergence and curl. For example, one of the equations states that the divergence of the magnetic field is always zero, implying no magnetic monopoles exist. Another equation involves the curl of the electric field, which is related to the changing magnetic field over time.
What is the difference between applying the gradient, divergence, and curl to fields in vector calculus?
-The gradient is applied to a scalar field and results in a vector field, representing the direction of steepest change. The divergence is applied to a vector field and results in a scalar field, representing how much the field spreads out from a point. The curl is applied to a vector field and results in another vector field, representing the rotation or circulation of the original field.
Outlines
🔍 Introduction to Differential Operators: Grad, Div, and Curl
In this paragraph, the speaker introduces three important differential operators—gradient, divergence, and curl—commonly abbreviated as grad, div, and curl. These operators are frequently used in physics and other fields. The speaker explains that the video will first clarify the meaning of each operator and then explore their applications.
🔺 Understanding the Nabla Operator (Del)
This paragraph introduces the nabla or del operator, represented by a downward-pointing triangle. Del acts as a vector in three dimensions with partial derivatives along the x, y, and z axes. It measures how a quantity changes over small distances in different directions. The speaker also notes that del is essential for understanding the grad, div, and curl operators and is part of vector calculus.
🌸 Example: Applying the Del Operator to a Flour Distribution
Using the example of a flour packet being squished, the speaker explains how the del operator is applied. The flour distribution, denoted as 'f,' is analyzed by taking the partial derivative with respect to x, dx. The resulting value indicates how quickly the distribution changes along the x-axis. The speaker emphasizes that partial derivatives represent changes in one direction while keeping other variables constant.
📏 Scalar Fields and Gradient of a Scalar Field
This section dives deeper into scalar fields, which assign values to points in space, such as altitude on a map. By applying the del operator to a scalar field, one obtains a vector field, which represents the rate and direction of the fastest change at each point. The speaker uses a height map to illustrate this concept, where vectors point in the direction of steepest ascent.
🌬️ Vector Fields: Gradient, Wind, and Electric Fields
Here, the speaker explains vector fields, which assign vectors (rather than scalar values) to each point in space. Examples include the direction of wind flow or the electric field created by charged particles. The divergence of a vector field is introduced, showing how it measures the net amount of field entering or leaving a region, with electric fields obeying specific rules from Maxwell's equations.
🔄 Curl: Measuring the Rotation of Vector Fields
The curl of a vector field is introduced as a measure of its circulation or rotation. Using a water flow example, the speaker explains how a fan dropped into flowing water would rotate, with the curl operator representing this rotation as a new vector field. The direction and magnitude of the resulting vector describe the axis and strength of the rotation.
🔗 Applications of Grad, Div, and Curl in Physics
This paragraph highlights the application of grad, div, and curl in physics. The gradient of a scalar field can represent a gravitational field, while Maxwell's equations link the divergence of the electric field and the curl of the magnetic field. These operators help explain intricate relationships between electric, magnetic, and gravitational fields.
👋 Conclusion and Call to Action
In the concluding paragraph, the speaker wraps up the video, thanking viewers for watching and encouraging them to like, subscribe, and support the channel through Patreon. The speaker also mentions other related videos for those interested in further exploration of the discussed topics.
Mindmap
Keywords
💡Gradient (Grad)
💡Divergence (Div)
💡Curl
💡Nabla (Del) Operator
💡Scalar Field
💡Vector Field
💡Partial Derivative
💡Dot Product
💡Cross Product
💡Maxwell's Equations
Highlights
Introduction of the key differential operators: Gradient, Divergence, and Curl, often shortened to Grad, Div, and Curl, which are essential in physics and other fields.
Explanation of the nabla (∇) or del operator, which acts as a vector with components representing partial derivatives in three dimensions (d/dx, d/dy, d/dz).
Clarification that ∇ can be applied to scalar or vector fields to measure the rate of change in different directions, linking vector calculus and physics.
Demonstration of gradient (grad) by applying ∇ to a scalar field to find how fast the quantity changes at each point, resulting in a vector field.
Explanation of divergence (div) as the dot product of ∇ and a vector field, measuring the net flow of a vector field, leading to a scalar field.
Illustration of how the divergence of the electric field shows whether the field originates from a source (positive charge) or a sink (negative charge).
Description of curl as the cross product of ∇ and a vector field, measuring the rotational motion or circulation at each point, resulting in another vector field.
Visualization of curl by imagining water flow and a fan placed in it, with the fan’s rotation representing the vector field’s curl.
Notion that scalar fields can represent real-world quantities like altitude or a distribution of flour in space, helping to visualize grad, div, and curl concepts.
Connection between grad and the gravitational field, with the gradient of gravitational potential φ resulting in the gravitational field, explained with negative sign convention.
Application of div in Maxwell’s equations, showing how the divergence of the magnetic field is always zero, indicating no isolated magnetic poles.
Curl’s role in Maxwell’s equations, where the curl of the electric field is linked to the time rate of change of the magnetic field, reinforcing the deep connection between electric and magnetic fields.
Explanation of scalar and vector fields: Scalar fields assign a single value at each point, while vector fields assign a direction and magnitude.
Visual example of the gradient of a scalar field using a height map, where the vectors point toward the direction of the steepest increase.
Final notes on the applications of grad, div, and curl in physics, particularly in electromagnetism and gravitational theory, illustrating the operators’ widespread use in theoretical and applied contexts.
Transcripts
hey everyone parth here and in this
video we will be looking at an
important group of differential
operators known as gradient divergence
and curl
these are often shortened to grad div
and curl and they're used very regularly
in the world of physics as well as
elsewhere
so we'll start by understanding what
each one of these actually means and
then we'll be looking at some
applications
to understand the grad div and curl
operators we need to start by thinking
about this downward pointing triangle
known as the nabla or del del can be
thought of as a vector
in three dimensions it looks something
like this the components of the vector
are partial derivative d by d
x partial derivative d by d y and
partial derivative d by d
z each one of these measures essentially
how quickly a particular quantity
changes
over a small distance in the x direction
y direction and z direction
now if you've seen my recent poisson
equation video and you're familiar with
partial derivatives as well as
del then feel free to skip to this
timestamp here so let's take a more
detailed look at dell
now for example if we've got a packet of
flour and we open this packet and we
decide to squish it
so the flour goes everywhere and then we
plot how much flour is found at every
point along
the x direction we can see that our
flour distribution would look something
like this lots of flower near the origin
and then less and less the further we
get away from the origin
let's also say that we label this flower
distribution as f
and let's now try and find df by dx this
simply measures how quickly our flower
distribution changes
as we move along the x-axis we can think
of this as measuring the gradient or
slope
of our flower distribution function over
here for example our flower distribution
drops off very quickly so the gradient
is steep
and hence df by dx has a large size and
of course is negative because the flower
is decreasing
whereas in this region the amount of
flower is not changing a huge amount
therefore the gradient is shallow and df
by dx has a small
magnitude and is again negative because
the flower is decreasing as we move from
left to right
basically d by dx of our flower
distribution f
is simply measuring how quickly the
flower distribution changes
now we could apply this nabla operator
to our function
f and we would get in the first instance
df by dx as we've already seen
except the nabla operator has these
weird curly d's
they're not normal d's now as it turns
out these curly d's are representing
what's known as a partial
derivative and luckily these are fairly
simple to think about at least in a
basic way
if we realize that our flower
distribution for example doesn't just
vary over the x direction
but it also varies in other directions
for example the y direction then we can
understand that the curly d's in df by
dx
mean that we're only measuring the
change in the x direction whilst keeping
everything else constant
similarly the curly df by dy isolates
out the change in the flower
distribution in the y direction
whilst keeping everything else constant
so we're not having to worry about the
change in the x direction
anyway so that's what dell ends up
representing and as we've already seen
this is what dell looks like in three
dimensions if we're only working in two
dimensions then del would look like this
and so on we'll notice that del is a
vector and it contains partial
derivatives which are studied in
calculus and hence del
is an operator in the area of
mathematics known as vector calculus but
we keep saying this word
operator what does it actually mean well
the nabla operator
can operate on or do stuff to certain
mathematical entities in this case
scalar fields or vector fields a scalar
field is basically a field of numbers or
values
in other words every point in a given
space whether that's real space or some
abstract space that we're considering
can be assigned a value
we can use scalar fields to represent
things like altitude on a map
in this particular case we're using it
to represent how high above sea level
you would be
if you were to stand on that point of
the earth or it could be used to
represent the amount of flour found in a
given region of space
after we'd squished our bag of flour and
of course those are just two of the more
physical examples see if you can come up
with your own example of how we can use
scalar fields
now the del operator can be used to find
out how quickly our scalar field changes
at
every point in other words we can find
the gradient of the scalar field
let's stick with our height above
c-level example from earlier
by applying the del operator to our
scalar field
h we get something like this the diagram
shows the gradient
of our scalar field h at each point we
see that there is an arrow or a vector
and each vector points in the direction
that the scalar field
increases most quickly so for example if
our scalar field looks something like
this
and we focus in on the point in the
middle then we can see that the scalar
field increases most quickly
in this direction so in our diagram of
the gradient of our field f
we would get a vector pointing in that
direction and of course the size or
magnitude of that vector
represents exactly how much our scalar
field is changing
so the crux of the matter is that
applying our nebula operator to a scalar
field
gives us a vector field that represents
the rate and direction
of fastest change of our original scalar
field
by the way a vector field is just a
field where we can assign a vector to
every point in that space
and in this case we can see that the
gradient of a scalar field will end up
being a vector field
however a vector field is not always
restricted to just being the gradient of
a scalar field
we can have other vector fields that
aren't necessarily the gradient of a
scalar field too
for example we can think of a vector
field that represents the direction in
which
wind is flowing the direction of each
vector tells us the direction in which
air particles are moving at that point
in space
and the magnitude or size of the vector
tells us how quickly they're moving the
speed of the particles
also in physics we can use a vector
field to represent the electric field
created by charged particles which
actually represents the following
if we were to take a small positively
charged particle and place it at a
particular point in the electric field
then the field lines tell us the size
and direction of the force
that that positive charge would
experience now here's something
interesting we can find out about every
vector field let's imagine that we think
of some imaginary sphere
in this region of space we can measure
exactly how much electric field
enters our sphere and equally we can
measure exactly how much electric field
leaves our imaginary sphere here we see
a certain number of
long arrows entering our sphere which
means that a certain amount of electric
field is entering our sphere or at least
we can imagine it that way
and on the other side we see a lot more
smaller
electric field lines leaving the sphere
and it turns out that for electric
fields specifically
it is always true that the amount of
electric field entering our imaginary
surface
has to equal the amount of electric
field leaving our imaginary surface
the amount coming in always has to
balance out exactly the amount going out
and this is always true except for when
our imaginary surface
is surrounding a charged particle a
positive charge is known as the source
of an electric field which means that
electric field lines originate from a
positive charge
and if we find a positive electric
charge within our imaginary sphere
then the net effect is that electric
field lines are said to be leaving
the sphere conversely a negative
electric charge is said to be a sink of
electric field lines in that electric
field lines end at negative charges
therefore if a negative charge is found
within our imaginary sphere
then the net effect is that electric
field lines are actually entering our
sphere
and as we've already seen in any other
region where there are no charged
particles within our sphere
the net effect is that the amount coming
in exactly cancels out the amount going
out
the reason that the electric field
behaves in this way is specifically
because of
this maxwell equation if you haven't
seen my video covering this maxwell
equation then check it out up here
now in this maxwell equation we can see
that the del operator
is indeed operating but it's no longer
acting as the gradient operator
and we can see that specifically because
of this little dot in between
the del and the e representing the
electric field
this dot is representing a dot product
or a scalar product
between del and the electric field e for
those of you not familiar with the dot
product between two vectors it's when
you take the corresponding components
from the two vectors
multiply them together and then add up
all these little products
but in this situation where the first
vector is the del
what we actually do is apply the partial
derivative to the corresponding
component of the electric field
and remember that the electric field is
a vector field and what we've just seen
is how to find
the divergence of our electric field
even though we see a del
in this location we're no longer taking
the gradient as we saw earlier with the
scalar field
we're now taking the divergence of the
vector field and that's all
changed simply by this little dot and
when we take the divergence off a vector
field what we end up with
is a scalar field in this particular
case it tells us exactly how much
electric field is entering or leaving a
particular point
and this is slightly different to the
gradient operator from earlier because
the gradient operator is applied to a
scalar field
and gives us a vector field now we've
seen how we can take a dot product
between del
and a vector field and as it turns out
we can also find the cross product
for a moment if we think about two
arbitrary vectors and we try and find
the cross product or the vector product
between them
the end result is usually a third vector
perpendicular to the first two
that is also a measure of how unaligned
the two vectors are and this is what i
mean by that
the size of the vector that we get by
taking the cross product between the
first two vectors
will be as large as possible if the
first two vectors are orthogonal to each
other or at right angles to each other
whereas if the two original vectors are
exactly aligned or exactly anti-aligned
then the vector resulting from the cross
product will have a size of zero
but if we now come back to thinking
about vector fields and del
we can find the cross product between
del and a vector field
and in this case things are slightly
different the cross product also known
as the curl
of the vector field is a measure of the
circulation
of our vector field let's imagine that
our vector field is representing some
sort of fluid flow let's say some water
flowing in like
a lake and in this water we drop some
sort of plastic fan
at every point in our vector field we
can measure the rotation that this fan
would experience
and whether it will rotate clockwise or
anti-clockwise
now we can represent the rotation of our
fan
with another vector that points along
the axis
of rotation a common convention is to
say that if the fan
rotates clockwise then our vector
representation will point
downward and if it rotates
anti-clockwise then it will point up
towards us
and the size or magnitude of this vector
represents the size or magnitude of the
rotation
and this is exactly what we measure when
we apply the curl operator
to our original vector field the end
result is another vector field that
represents the axis of rotation
of each part of the original fluid flow
in this case
it's important to note though that the
rotation is not a property of the fan
it's actually the fluid flow that's
causing our imaginary fan to rotate
and therefore the curl is actually
measuring something about the original
vector field
and so bringing it all the way around we
earlier saw that when we apply gradient
to a scalar field
the end result is a vector field when we
apply divergence to a vector field the
end result is a scalar field
and when we apply curl to a vector field
the end result is a vector field
but it's also important to understand
what each one of these represents
now in physics all three of these
operators are used very regularly
for example the gradient of a scalar
field phi
is directly related to the gravitational
field produced by
some object with mass and in this
particular case the scalar field phi is
known as the gravitational potential
now in this case we've got a negative
sign because the rate of increase of the
gravitational potential
points in the opposite direction to the
gravitational field itself but we can
see the use of the grand operator on a
scalar field in this particular instance
in electromagnetism we've seen that
maxwell's equations deal with electric
and magnetic fields
and we've already seen one of these
equations earlier in the video
but another one tells us that if our
theory of classical electromagnetism is
correct
then the divergence of the magnetic
field is always
zero in other words there are no lone
sources or sinks
of the magnetic field for more
information on this check out my video
on that maxwell equation and finally
another maxwell equation
tells us that the curl of the electric
field is equal to minus the time rate of
change of the magnetic field
in this way we see an intricate link
between electric and magnetic fields and
we see also the use of the curl operator
for more information on that maxwell
equation check out this video up here
and with all of that being said i'm
going to finish up here thank you so
much for watching if you enjoyed the
video please do hit the thumbs up button
and subscribe for more fun physics
content
hit the bell button if you'd like to be
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thank you as always for your wonderful
support and i will see you very soon
[Music]
you
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