Gradient & Related Calculations
Example
:
We are out recording measurements of some scalar quantity Q and after many observations of this
quantity (in both space and time) we arrive at the following family of curves (for different values
of Q):
a.) Calculate the gradient of Q
.
From our analytic expression above we have:
Thus the gradient of Q is given by the Cartesian vector
Applying graphical vector analysis, we can plot the gradient (it is -1 in the x-direction, and -1 in
the y direction) vector (does it matter where I put it, i.e. have students tell me that the gradient is
uniform, therefore I can plop it down anywhere on Fig. 1 above - of course this won’t always be
the case!).
b.)What is the tangent vector to the isolines?
We can apply the following
where
i
1
=
i
, and
i
2
=
j
, and dx
1
=dx, dx
2
=dy, and dx
3
=dz=0 (xy plane only). So we need to calcu-
late the slopes dx/ds, dy/ds. These slopes are just the limit as the difference between the two posi-
tion vectors
∆
r
(=
∆
s
t
)approaches zero (recall that the slope is valid at a ‘pt’). Imagine the two
points on the isoline - green pts in Fig.1, approaching one another and as they do, the triangle that
constitutes the components of the vector drawn along the isoline, shrinks in area. Note that both
y
2
x
Q x y
,
(
)
∴
–
2
x
–
y
–
=
=
∇
Q
i
ˆ
∂
∂
x
-----
j
ˆ
∂
∂
y
-----
k
ˆ
∂
∂
z
-----
+
+
Q
(
)
i
ˆ
∂
Q
∂
x
-------
j
ˆ
∂
Q
∂
y
-------
k
ˆ
∂
Q
∂
z
-------
+
+
=
=
∂
Q
∂
x
-------
1,
∂
Q
∂
y
-------
–
1,
and
∂
Q
∂
z
-------
–
0
=
=
=
∇
Q
i
ˆ
–
j
ˆ
–
=
t
ˆ
i
ˆ
1
dx
1
ds
-------
i
ˆ
2
dx
2
ds
-------
i
ˆ
3
dx
3
ds
-------
+
+
=
x
y
1
2
3
4
1
4
3
2
Q = 1
Q = 0
Q = -1
Q = -2
i
ˆ
–
j
ˆ
–
∇
Q
0
r
1
r
2
Fig. 1
∆
s
∆
x
∆
y

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