03.2 Change one reason at a time

Content and key concepts

On the previous page, sensitivity to minor change.
If the result depends on two numbers, first we change one with a fixed friend.
f=x2+3y2; point [2.1].
Координаты и расчёт геометрического примера приведены в тексте. Для изменения пространственного рисунка включитеJavaScript.
One coordinate experiment at a time
The function of several variables has several independent small motion modes.
f
(
x
,
y
)
=
x
2
+
3
y
2
f(x,y)=x^2+3y^2
f
(
x
,
y
)
=
x
2
+
3
y
2
Private derivative for
x
x
x
defined with fixed
y
y
y
:
∂
f
∂
x
(
x
,
y
)
=
lim
⁡
h
→
0
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
h
=
2
x
.
\frac{\partial f}{\partial x}(x,y)
=\lim_{h\to0}\frac{f(x+h,y)-f(x,y)}h=2x.
∂
x
∂
f
​
(
x
,
y
)
=
h
→
0
lim
​
h
f
(
x
+
h
,
y
)
−
f
(
x
,
y
)
​
=
2
x
.
Same with fixed
x
x
x
:
∂
f
∂
y
(
x
,
y
)
=
6
y
.
\frac{\partial f}{\partial y}(x,y)=6y.
∂
y
∂
f
​
(
x
,
y
)
=
6
y
.
Point
(
2
,
1
)
(2,1)
(
2
,
1
)
Yes, we do.
f
=
7
f=7
f
=
7
,
∂
f
/
∂
x
=
4
\partial f/\partial x=4
∂
f
/
∂
x
=
4
,
∂
f
/
∂
y
=
6
\partial f/\partial y=6
∂
f
/
∂
y
=
6
The first number is the section slope
f
(
x
,
1
)
=
x
2
+
3
f(x,1)=x^2+3
f
(
x
,
1
)
=
x
2
+
3
, the second is the slope of another section
f
(
2
,
y
)
=
4
+
3
y
2
f(2,y)=4+3y^2
f
(
2
,
y
)
=
4
+
3
y
2
These are not two competing derivatives of the same line, but two co-ordinates on the same surface.
Full small increment gathers all coordinates
Prix
δ
=
[
δ
x
,
δ
y
]
T
\delta=[\delta_x,\delta_y]^T
δ
=
[
δ
x
​
,
δ
y
​
]
T
the first order of change is
Δ
f
l
i
n
=
∂
f
∂
x
δ
x
+
∂
f
∂
y
δ
y
.
\Delta f_{\mathrm{lin}}
=\frac{\partial f}{\partial x}\delta_x
+\frac{\partial f}{\partial y}\delta_y.
Δ
f
lin
​
=
∂
x
∂
f
​
δ
x
​
+
∂
y
∂
f
​
δ
y
​
.
Point
(
2
,
1
)
(2,1)
(
2
,
1
)
and to the
δ
=
[
0.01
,
0.02
]
T
\delta=[0.01,0.02]^T
δ
=
[
0.01
,
0.02
]
T
:
Δ
f
l
i
n
=
4
⋅
0.01
+
6
⋅
0.02
=
0.16.
\Delta f_{\mathrm{lin}}=4\cdot0.01+6\cdot0.02=0.16.
Δ
f
lin
​
=
4
⋅
0.01
+
6
⋅
0.02
=
0.16.
Exact change equals
f
(
2.01
,
1.02
)
−
f
(
2
,
1
)
=
0.1613.
f(2.01,1.02)-f(2,1)=0.1613.
f
(
2.01
,
1.02
)
−
f
(
2
,
1
)
=
0.1613.
Balance
0.0013
0.0013
0.0013
can be checked without rounded geometry:
0.01
2
+
3
(
0.02
)
2
=
0.0013.
0.01^2+3(0.02)^2=0.0013.
0.0
1
2
+
3
(
0.02
)
2
=
0.0013.
Each private derivative takes account of its linear contribution, and the mixed movement builds these deposits.
"The rest is fixed" is a condition for determining
The phrase does not claim that the signs of real dataset are statistically independent. It sets up a virtual local experiment over a function. If accessible dots are limited to curves
y
=
x
y=x
y
=
x
the actual movement on this curve changes both coordinates:
ϕ
(
t
)
=
f
(
t
,
t
)
=
4
t
2
,
ϕ
′
(
t
)
=
8
t
.
\phi(t)=f(t,t)=4t^2,
\qquad \phi'(t)=8t.
ϕ
(
t
)
=
f
(
t
,
t
)
=
4
t
2
,
ϕ
′
(
t
)
=
8
t
.
Point
t
=
2
t=2
t
=
2
The same number gives chain rule:
d
d
t
f
(
x
(
t
)
,
y
(
t
)
)
=
∂
f
∂
x
d
x
d
t
+
∂
f
∂
y
d
y
d
t
=
4
+
12
=
16.
\frac{d}{dt}f(x(t),y(t))
=\frac{\partial f}{\partial x}\frac{dx}{dt}
+\frac{\partial f}{\partial y}\frac{dy}{dt}
=4+12=16.
d
t
d
​
f
(
x
(
t
)
,
y
(
t
))
=
∂
x
∂
f
​
d
t
d
x
​
+
∂
y
∂
f
​
d
t
d
y
​
=
4
+
12
=
16.
You can't just leave
∂
f
/
∂
x
\partial f/\partial x
∂
f
/
∂
x
if
y
y
y
Depends, too.
t
t
t
Conversely, the relationship in the data does not change the definition of the private derivative of the function itself.
When the coefficient 3 is replaced by the factor 3
b
b
b
receive another function
f
b
=
x
2
+
b
y
2
f_b=x^2+by^2
f
b
​
=
x
2
+
b
y
2
Then.
∂
f
b
∂
y
=
2
b
y
.
\frac{\partial f_b}{\partial y}=2by.
∂
y
∂
f
b
​
​
=
2
b
y
.
The surface, the length of the level lines and the sensitivity of the
y
y
y
This is a model change, not a shift on the old surface.
x, y = 2, 1
dx, dy = .01, .02
linear = 2*x*dx + 6*y*dy
exact = (x + dx)**2 + 3*(y + dy)**2 - (x*x + 3*y*y)
assert abs(linear - .16) < 1e-12
assert abs(exact - .1613) < 1e-12
The code repeats the operations just taken care of.
Completely checked Python-example Chapter
Available with a separate file; the short piece shown does not replace its implementation.
Each reason is first changed separately; the Gradient combines sensitivity; a small step backwards has reduced the value in this example.
How does this question go about the result?
Python examples of Chapter 03
Academic base