Engineering Mathematics Exam > Engineering Mathematics Videos > Gradient (Engineering Mathematics)
Gradient (Engineering ) Video Lecture - Engineering
Text Transcript from Video
Let's learn a little bit about
the gradient, and we'll use the
same function we've been using
since we're pretty familiar
with it's graph and it's
partial derivatives.
So let's say that f of x y is
equal to x squared plus x
times y plus y squared.
Now we're going to take the
gradient of this, and then I'll
give you a little bit of
intuition on what
a gradient is.
We're going to take the
gradient in two-dimensional
space, and you'll see what
that means in a second.
You can extend this
to any dimension.
So the gradient that we use
this vector differential
operator-- this upside down
triangle-- the gradient of f is
equal to the partial of f in
the x direction times the unit
vector i-- which is the unit
vector in the x direction so it
gives the magnitude of the
slope and the x direction, and
then it points the vector in
that direction-- plus the
partial with respect to
y in the j direction.
And obviously if there's a
function of more variables you
could keep going and you'd
multiply the magnitude of the
derivative in each of those
dimension's directions times
the unit vector in
that dimension.
Well what does this mean?
And another way to view it, you
could have also written this as
the gradient of f is equal to
the partial with respect to x
of f of x y and then all of
that in the i direction.
This is just
different notation.
Plus the partial with respect
to y times f of x y.
Not times, operated
on f of x y.
So the partial of f of x y
with respect to y times
the j unit vector.
And because of this notation, a
lot of people view this delta
operator-- and actually it's
pretty consistent when we learn
about the divergence and the
curl-- a lot of people view
this delta operator, they
actually define it.
They say that is equal to the
partial with respect to x in
the i direction plus the
partial with respect to
y in the j direction.
And then if you wanted to do it
into three space-- that's not
going to apply to our problem
because we only care about
two dimensions right now.
And you'll see what I mean.
Because even though we're
applying it to a
three-dimensional surface, but
then you can do the partial
with respect to z
times the k vector.
And then you could extend it to
n dimensions, but it becomes
very hard to visualize.
Actually it becomes hard
to visualize beyond what
we're doing right now.
So what does this mean?
Well before I go into what it
means, let's actually calculate
it for this and then I'll
show you what it means.
Let's actually calculate
the gradient of f.
So the gradient of our
particular function, it's a
partial with respect to x.
So we figure that out.
That's 2x plus y, and then
that doesn't matter anymore.
So it's 2x plus y.
That's the partial of this
function with respect to x.
And we multiply that in
the i direction, or
in the x direction.
Plus the partial with respect
to y, and this is 2y plus x.
And we calculated this in
the two videos on the
partial derivatives.
And we multiply that
in the j direction.
So what does that look like?
Well what are each of the
components of this vector?
What is 2x plus y in
the i direction?
What does that
vector look like?
What is 2y plus x in the
j direction look like?
And this I have graphed.
So this is the same surface
we've been working with, but
now I've plotted these
points right here.
These are just points that the
software has picked to actually
display these vectors.
And if you look at it,
this is the x-axis.
And I can rotate this.
I can pull it down
a little bit.
And then I can spin it around.
I think that's pretty neat.
But anyway.
This parallel to this
line is the vector in
the x-axis, right?
So at this point I've actually
calculated the gradient.
This vector says, what is the
magnitude of this vector is the
partial derivative of the
function, or the partial
derivative of the surface,
or the partial derivative
of z with respect to x.
And it's direction, it just
goes in the x direction,
because it's that times
the i unit vector.
So this vector is the partial
derivative of z with respect
to x times the i unit
vector at that point.
So we just calculated
the partial derivative
at that point.
It gave this length.
And then the direction is just
the i unit vector, or in the
direction of increasing x.
Similarly, this vector right
here-- I hope you can see it.
Let me see if I can actually
zoom in a little bit.
That might be useful.
Zoom in, there you go.
And I wanted to see the axes,
which you can't see now that
I'm zoomed in, but you took my
word this was the x direction.
Now this is the y direction,
or the same direction
as the j unit vector.
The j unit vector goes in
the same direction as y.
And the magnitude is determined
by the partial derivative of z
with respect to y
at that point.
And that's the magnitude.
And we saw there was some
symmetry, so the magnitude of
this vector is the same as the
magnitude of this vector.
And then when you add the
two vectors together,
you get this vector.
And notice that none of these
vectors have any dimensions
in the z dimension.
They're all kind of giving you
directions in the x y plane.
And why is that interesting?
Well the gradient-- and this is
the intuition-- the gradient
tells you the direction in the
x y plane you should travel
in order to get the maximum
slope in the z dimension.
Or another way to view it.
Remember the partial derivative
with respect to x said what is
the slope in the x direction.
The partial derivative with
respect to y said what is the
slope in the y direction.
But you could take the partial
derivative with respect to any
direction, and the gradient
gives you the direction in
which the slope is the largest.
So let me zoom out a little bit
because I want you to see the
actual axes There's the axes So
all this says if I were to
go in this direction, I
get the maximum slope.
So if I go in this direction,
my z goes up like that.
Let me see if I can rotate
this a little bit.
I don't want to
scale it anymore.
Let me do the rotated.
See if I can show that to you.
So if I go in that direction in
the x y plane, I get maximum z.
If I go in that direction I
get a maximum upward slope.
That's what the gradient tells
you, how do you get the
maximum upward slope.
And if you were to take any
closed line where the z is
constant-- well actually I
don't want to get into that too
much, but this gradient will
actually be normal to any, or
it'll be perpendicular to any
curve where z is constant.
I don't want to get too
involved with that right
now, so let's go back here.
Maybe it's more clear if we
look from below the graph.
So if you go to this point
right here, this is the
magnitude of this vector,
shows what is the slope
in the x direction.
The magnitude of this vector
is what is the slope
in the y direction.
And then when you add them
together, you get the gradient.
And the gradient says, well, if
I travel in this direction in
the x y plane-- notice that
none of these have any z's.
The plane defined by all
of these vectors is all
flat in the z dimension.
But if I would have traveled
this direction in the x y
plane, then I will get the
maximum increase in z
per unit that I travel.
And let's actually show
you what it looks like
in the x y plane.
So if I just go head on-- so
I'm above the graph looking
straight down at the graph, and
then the colors just show kind
of you know where we are-- if I
travel in this direction in
the x y plane, I get my
maximum increase in z.
If I'm here-- notice here the x
component of the gradient is
much larger than
the y component.
So I need to travel a little
bit more in the x direction,
and I'll get the maximum change
in z if I travel there.
Another way to think about it.
If I'm on a hill, the gradient
of that surface will tell you
at any point what direction you
need to travel in to go
up the hill fastest.
Or the direction which
the steepness of the
hill is maximum.
This is more of a bowl as
opposed to a hill, but anyway.
I'll just rotate it around.
Hopefully that
makes some sense.
And I want you to think
about it a little bit more.
And we'll do a few more
problems where we just
calculate gradients, just
because I think it's useful
to get the mechanics.
But in my opinion at least the
intuition is a little bit
harder to get your head around
than the actual mechanics,
but once you get it, it
makes a lot of sense.
I will see you in
the next video.
the gradient, and we'll use the
same function we've been using
since we're pretty familiar
with it's graph and it's
partial derivatives.
So let's say that f of x y is
equal to x squared plus x
times y plus y squared.
Now we're going to take the
gradient of this, and then I'll
give you a little bit of
intuition on what
a gradient is.
We're going to take the
gradient in two-dimensional
space, and you'll see what
that means in a second.
You can extend this
to any dimension.
So the gradient that we use
this vector differential
operator-- this upside down
triangle-- the gradient of f is
equal to the partial of f in
the x direction times the unit
vector i-- which is the unit
vector in the x direction so it
gives the magnitude of the
slope and the x direction, and
then it points the vector in
that direction-- plus the
partial with respect to
y in the j direction.
And obviously if there's a
function of more variables you
could keep going and you'd
multiply the magnitude of the
derivative in each of those
dimension's directions times
the unit vector in
that dimension.
Well what does this mean?
And another way to view it, you
could have also written this as
the gradient of f is equal to
the partial with respect to x
of f of x y and then all of
that in the i direction.
This is just
different notation.
Plus the partial with respect
to y times f of x y.
Not times, operated
on f of x y.
So the partial of f of x y
with respect to y times
the j unit vector.
And because of this notation, a
lot of people view this delta
operator-- and actually it's
pretty consistent when we learn
about the divergence and the
curl-- a lot of people view
this delta operator, they
actually define it.
They say that is equal to the
partial with respect to x in
the i direction plus the
partial with respect to
y in the j direction.
And then if you wanted to do it
into three space-- that's not
going to apply to our problem
because we only care about
two dimensions right now.
And you'll see what I mean.
Because even though we're
applying it to a
three-dimensional surface, but
then you can do the partial
with respect to z
times the k vector.
And then you could extend it to
n dimensions, but it becomes
very hard to visualize.
Actually it becomes hard
to visualize beyond what
we're doing right now.
So what does this mean?
Well before I go into what it
means, let's actually calculate
it for this and then I'll
show you what it means.
Let's actually calculate
the gradient of f.
So the gradient of our
particular function, it's a
partial with respect to x.
So we figure that out.
That's 2x plus y, and then
that doesn't matter anymore.
So it's 2x plus y.
That's the partial of this
function with respect to x.
And we multiply that in
the i direction, or
in the x direction.
Plus the partial with respect
to y, and this is 2y plus x.
And we calculated this in
the two videos on the
partial derivatives.
And we multiply that
in the j direction.
So what does that look like?
Well what are each of the
components of this vector?
What is 2x plus y in
the i direction?
What does that
vector look like?
What is 2y plus x in the
j direction look like?
And this I have graphed.
So this is the same surface
we've been working with, but
now I've plotted these
points right here.
These are just points that the
software has picked to actually
display these vectors.
And if you look at it,
this is the x-axis.
And I can rotate this.
I can pull it down
a little bit.
And then I can spin it around.
I think that's pretty neat.
But anyway.
This parallel to this
line is the vector in
the x-axis, right?
So at this point I've actually
calculated the gradient.
This vector says, what is the
magnitude of this vector is the
partial derivative of the
function, or the partial
derivative of the surface,
or the partial derivative
of z with respect to x.
And it's direction, it just
goes in the x direction,
because it's that times
the i unit vector.
So this vector is the partial
derivative of z with respect
to x times the i unit
vector at that point.
So we just calculated
the partial derivative
at that point.
It gave this length.
And then the direction is just
the i unit vector, or in the
direction of increasing x.
Similarly, this vector right
here-- I hope you can see it.
Let me see if I can actually
zoom in a little bit.
That might be useful.
Zoom in, there you go.
And I wanted to see the axes,
which you can't see now that
I'm zoomed in, but you took my
word this was the x direction.
Now this is the y direction,
or the same direction
as the j unit vector.
The j unit vector goes in
the same direction as y.
And the magnitude is determined
by the partial derivative of z
with respect to y
at that point.
And that's the magnitude.
And we saw there was some
symmetry, so the magnitude of
this vector is the same as the
magnitude of this vector.
And then when you add the
two vectors together,
you get this vector.
And notice that none of these
vectors have any dimensions
in the z dimension.
They're all kind of giving you
directions in the x y plane.
And why is that interesting?
Well the gradient-- and this is
the intuition-- the gradient
tells you the direction in the
x y plane you should travel
in order to get the maximum
slope in the z dimension.
Or another way to view it.
Remember the partial derivative
with respect to x said what is
the slope in the x direction.
The partial derivative with
respect to y said what is the
slope in the y direction.
But you could take the partial
derivative with respect to any
direction, and the gradient
gives you the direction in
which the slope is the largest.
So let me zoom out a little bit
because I want you to see the
actual axes There's the axes So
all this says if I were to
go in this direction, I
get the maximum slope.
So if I go in this direction,
my z goes up like that.
Let me see if I can rotate
this a little bit.
I don't want to
scale it anymore.
Let me do the rotated.
See if I can show that to you.
So if I go in that direction in
the x y plane, I get maximum z.
If I go in that direction I
get a maximum upward slope.
That's what the gradient tells
you, how do you get the
maximum upward slope.
And if you were to take any
closed line where the z is
constant-- well actually I
don't want to get into that too
much, but this gradient will
actually be normal to any, or
it'll be perpendicular to any
curve where z is constant.
I don't want to get too
involved with that right
now, so let's go back here.
Maybe it's more clear if we
look from below the graph.
So if you go to this point
right here, this is the
magnitude of this vector,
shows what is the slope
in the x direction.
The magnitude of this vector
is what is the slope
in the y direction.
And then when you add them
together, you get the gradient.
And the gradient says, well, if
I travel in this direction in
the x y plane-- notice that
none of these have any z's.
The plane defined by all
of these vectors is all
flat in the z dimension.
But if I would have traveled
this direction in the x y
plane, then I will get the
maximum increase in z
per unit that I travel.
And let's actually show
you what it looks like
in the x y plane.
So if I just go head on-- so
I'm above the graph looking
straight down at the graph, and
then the colors just show kind
of you know where we are-- if I
travel in this direction in
the x y plane, I get my
maximum increase in z.
If I'm here-- notice here the x
component of the gradient is
much larger than
the y component.
So I need to travel a little
bit more in the x direction,
and I'll get the maximum change
in z if I travel there.
Another way to think about it.
If I'm on a hill, the gradient
of that surface will tell you
at any point what direction you
need to travel in to go
up the hill fastest.
Or the direction which
the steepness of the
hill is maximum.
This is more of a bowl as
opposed to a hill, but anyway.
I'll just rotate it around.
Hopefully that
makes some sense.
And I want you to think
about it a little bit more.
And we'll do a few more
problems where we just
calculate gradients, just
because I think it's useful
to get the mechanics.
But in my opinion at least the
intuition is a little bit
harder to get your head around
than the actual mechanics,
but once you get it, it
makes a lot of sense.
I will see you in
the next video.
About this Video
Apr 22, 2026 Last updated
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