}}).
87 |
88 | - When `$\domain(\theta) \subset \real^2$` is a *2D region* with `$\mu$` being the area measure, the extended boundary `$\boundary(\theta)$` comprises a set of curves (as illustrated in [the figure above](#fig_domain)) with `$\dmu$` being the curve-length measure.
89 |
90 | - When `$\domain(\theta) \subseteq \sph$` is a *curved 2D region* within the surface `$\sph$` of the unit sphere and `$\mu$` the solid-angle measure, the extended boundary `$\boundary(\theta)$` comprises a set of *spherical curves* with `$\dmu$` being the curve-length measure.
91 |
92 | - When `$\domain(\theta) \subset \real^3$` is a *3D volume* and `$\mu$` the volume measure, the extended boundary `$\boundary(\theta)$` becomes *2D surfaces* with `$\dmu$` being the surface-area measure.
93 |
94 | ### Definition of `$\Delta f$`
95 |
96 | In Eq. \eqref{eqn:dI_reynolds}, another important component of the *boundary* integral is `$\Delta f(\bx, \theta)$` that captures the difference in `$f(\bx, \theta)$` across a discontinuity boundary at `$\bx$`.
97 |
98 | Precisely, let `$\bx_{\bot}(\epsilon)$` be a regular curve parameterized by `$\epsilon$` (illustrated as the gray dashed line in the [figure above](#fig_domain)) that resides inside the domain `$\domain(\theta)$` and satisfies $\bx_{\bot}(0) = \bx$.
99 | When the domain `$\domain(\theta)$` is "flat", for example, we have
100 |
101 |
102 | $$
103 | \bx_{\bot}(\epsilon) := \bx + \epsilon\,\bn(\bx),
104 | $$
105 |
106 |
107 | where `$\bn(\bx)$` is the unit normal of the extended boundary at `$\bx$`.
108 | Then,
109 |
110 |
111 | \begin{equation}
112 | \label{eqn:delta_f}
113 | \Delta f(\bx, \theta) = \lim_{\epsilon \uparrow 0} f(\bx_{\bot}(\epsilon), \,\theta)
114 | - \lim_{\epsilon \downarrow 0} f(\bx_{\bot}(\epsilon), \,\theta).
115 | \end{equation}
116 |
117 |
118 |
119 | ### Normal Change Rate `$\vel$`
120 |
121 | Another key component of the *boundary* integral in Eq. \eqref{eqn:dI_reynolds} is the **normal change rate** term `$\vel$` which, at a high level, captures how fast the extended boundary `$\boundary(\theta)$` evolves along the normal `$\bn$` (with respect to `$\theta$`).
122 | Precisely, for any `$\bx \in \boundary(\theta)$`, we have
123 |
124 |
125 | \begin{equation}
126 | \label{eqn:vel}
127 | \vel(\bx) = \frac{\D\bx}{\D\theta} \cdot \bn(\bx),
128 | \end{equation}
129 |
130 |
131 | where "`$\cdot$`" denotes the dot (or scalar) product of two vectors.
132 |
133 | #### Parameterizing Discontinuity Boundaries
134 |
135 | Evaluating the derivative `$\D\bx/\D\theta$` on the right-hand side of Eq. \eqref{eqn:vel} requires the extended boundary `$\boundary(\theta)$` to be *locally parameterized*.
136 | Specifically, for each `$\theta$` and `$\bx \in \boundary(\theta)$`, we assume without loss of generality that `$\alphaBnd(\boldsymbol{\xi}, \theta)$` is a differentiable and bijective function satisfying that:
137 |
138 | - `$\alphaBnd(\cdot, \theta)$` maps an open ball centered at the origin `$\boldsymbol{0}$` to an open neighborhood around `$\bx$` within the extended boundary `$\boundary(\theta)$`;
139 |
140 | - `$\alphaBnd(\boldsymbol{0}, \theta) = \bx$`.
141 |
142 | Then,
143 |
144 |
145 | \begin{equation}
146 | \label{eqn:vel_raw}
147 | \frac{\D\bx}{\D\theta} = \frac{\D}{\D\theta} \alphaBnd(\boldsymbol{0}, \theta).
148 | \end{equation}
149 |
150 |
151 |
152 | #### Parameterization Independency
153 |
154 | The derivative `$\D\bx/\D\theta$` defined in Eq. \eqref{eqn:vel_raw} is *parameterization-dependent*.
155 | That is, it depends on the choice of the local parameterization specified by the function $\alphaBnd$.
156 |
157 | On the contrary, the normal change rate `$\vel(\bx)$` from Eq. \eqref{eqn:vel} is known to be *parameterization-independent*.
158 | In other words, regardless to the choice of $\alphaBnd$, `$\vel(\bx)$` remains identical.
159 |
160 |
165 |
--------------------------------------------------------------------------------
/manim/diff-render/basics/diff-int-1d/main.py:
--------------------------------------------------------------------------------
1 | from manim import *
2 | from manim.utils.color import Colors
3 | import numpy as np
4 |
5 | config.background_color = WHITE
6 |
7 | def init_axis(x_range, y_range, length=1., precision=0):
8 | ax = Axes(
9 | x_range = x_range,
10 | y_range = y_range,
11 | x_length = length*(x_range[1] - x_range[0]),
12 | y_length = length*(y_range[1] - y_range[0]),
13 | tips = False,
14 |
15 | axis_config = {
16 | "color": Colors.light_gray.value,
17 | # "include_numbers": True,
18 | "include_ticks": True,
19 | "decimal_number_config": {
20 | "num_decimal_places": precision,
21 | "color": Colors.light_gray.value,
22 | },
23 | "font_size": 32
24 | },
25 |
26 | x_axis_config = {
27 | "include_numbers": True,
28 | "numbers_to_include": np.arange(0., 1.01, .2),
29 | },
30 | )
31 | labels = ax.get_axis_labels(x_label="x", y_label="f").set_color(Colors.light_gray.value)
32 | return ax, labels
33 |
34 |
35 | class Example_Base(Scene):
36 | def __init__(self):
37 | super().__init__()
38 | self.ax, labels = init_axis((-.1, 1.2, .1), (-.1, .6, .1), length=10., precision=1)
39 | self.add(self.ax, labels)
40 |
41 | def get_plot(self, func, clr, theta_txt, dir):
42 | graph = self.ax.plot(func, x_range=[0., 1.], color=clr)
43 | label = self.ax.get_graph_label(graph, x_val=1., direction=dir, label="f(x, " + theta_txt + ")").scale(.8)
44 | area = self.ax.get_area(graph, x_range=[0., 1.], color=clr, opacity=.3).set_stroke(opacity=0.)
45 | area_label = MathTex("I(" + theta_txt + ")", color=clr).move_to(self.ax.c2p(.8, .4*func(.8)))
46 | return graph, label, area, area_label
47 |
48 | def get_piecewise_plot(self, x0, y0, y1, clr, theta_txt):
49 | graphL = self.ax.plot(lambda x: y0, x_range=[0., x0], color=clr)
50 | areaL = self.ax.get_area(graphL, x_range=[0, x0], color=clr, opacity=.3).set_stroke(opacity=0.)
51 | graphR = self.ax.plot(lambda x: y1, x_range=[x0, 1.], color=clr)
52 | areaR = self.ax.get_area(graphR, x_range=[x0, 1.], color=clr, opacity=.3).set_stroke(opacity=0.)
53 | area_label = MathTex("I(" + theta_txt + ")", color=clr).move_to(self.ax.c2p(.5, .5*min(y0, y1)))
54 | line = DashedLine(self.ax.c2p(x0, y0), self.ax.c2p(x0, y1), color=clr)
55 | return VGroup(graphL, graphR, line), VGroup(areaL, areaR), area_label
56 |
57 | class SuccessExample_0(Example_Base):
58 | def __init__(self):
59 | super().__init__()
60 | self.t0 = .35
61 | self.f0 = lambda x: self.t0*x*x
62 |
63 | def construct(self):
64 | graph0, label0, area0, area0_label = self.get_plot(self.f0, BLUE, r"\theta_0", UR*.25)
65 | self.add(graph0, label0, area0, area0_label)
66 |
67 |
68 | class SuccessExample_1(SuccessExample_0):
69 | def __init__(self):
70 | super().__init__()
71 | self.dt = .15
72 | self.f1 = lambda x: (self.t0 + self.dt)*x*x
73 |
74 | def construct(self):
75 | graph0, label0, _, _ = self.get_plot(self.f0, BLUE, r"\theta_0", UR*.25)
76 | graph0.set_stroke(opacity = .5)
77 | self.add(DashedVMobject(graph0, num_dashes=30), label0)
78 |
79 | graph1, label1, area1, area1_label = self.get_plot(self.f1, RED, r"\theta_0 + \Delta\theta", UR*.25)
80 | self.add(graph1, label1, area1, area1_label)
81 |
82 |
83 | class SuccessExample_2(SuccessExample_1):
84 | def __init__(self):
85 | super().__init__()
86 |
87 | def construct(self):
88 | graph0, label0, _, _ = self.get_plot(self.f0, BLUE, r"\theta_0", UR*.25)
89 | graph1, label1, _, _ = self.get_plot(self.f1, RED, r"\theta_0 + \Delta\theta", UR*.25)
90 | d_area = self.ax.get_area(graph0, x_range=[0., 1.], bounded_graph=graph1, color=ORANGE, opacity=0.3).set_stroke(opacity = 0.)
91 | self.add(d_area, graph0, label0, graph1, label1)
92 |
93 | arrow_x = .675
94 | arrow_y = .5*(self.f0(arrow_x) + self.f1(arrow_x))
95 | arrow = Arrow(self.ax.c2p(arrow_x - .2, arrow_y), self.ax.c2p(arrow_x, arrow_y), buff=0., color=ORANGE, max_tip_length_to_length_ratio=.1)
96 | self.add(arrow)
97 |
98 | d_area_label = MathTex(r"I(\theta_0 + \Delta\theta) - I(\theta_0)", color=ORANGE)
99 | d_area_label.scale(.8).next_to(arrow.get_start(), direction=LEFT)
100 | self.add(d_area_label)
101 |
102 | x0 = .9
103 | line = self.ax.get_vertical_line(self.ax.i2gp(x0, graph1), color=GRAY)
104 | self.add(line)
105 |
106 | dot0 = Dot(self.ax.c2p(x0, 0.), color=GRAY).scale(.8)
107 | dot1 = Dot(self.ax.c2p(x0, self.f0(x0)), color=BLUE).scale(.8)
108 | dot2 = Dot(self.ax.c2p(x0, self.f1(x0)), color=RED).scale(.8)
109 | self.add(dot0, dot1, dot2)
110 |
111 | dot0_label = MathTex(r"x", color=GRAY).scale(.8).next_to(dot0, direction=.5*DOWN)
112 | self.add(dot0_label)
113 |
114 | brace = Brace(Line(dot1.get_center(), dot2.get_center()), direction=LEFT, color=BLACK, fill_opacity=.5)
115 | brace_tex = MathTex(r"f(x, \theta_0 + \Delta\theta) - f(x, \theta_0)\\ \approx \left[\frac{\mathrm{d}}{\mathrm{d}\theta} f(x, \theta)\right]_{\theta = \theta_0} \,\Delta\theta", color=BLACK).scale(.8).set_opacity(.5)
116 | brace_tex.next_to(brace, LEFT)
117 | self.add(brace, brace_tex)
118 |
119 |
120 | class FailureExample_0(Example_Base):
121 | def __init__(self):
122 | super().__init__()
123 | self.t0 = .5
124 | self.y0 = .5
125 |
126 | def construct(self):
127 | graph0, area0, area0_label = self.get_piecewise_plot(self.t0, self.y0, .5*self.y0, BLUE, r"\theta_0")
128 | label0 = MathTex(r"f(x, \theta_0)", color=BLUE).scale(.8).next_to(self.ax.c2p(self.t0, .75*self.y0), direction=RIGHT)
129 | self.add(graph0, label0, area0, area0_label)
130 |
131 |
132 | class FailureExample_1(FailureExample_0):
133 | def __init__(self):
134 | super().__init__()
135 | self.dt = .1
136 |
137 | def construct(self):
138 | graph0, _, _ = self.get_piecewise_plot(self.t0, self.y0, .5*self.y0, BLUE, r"\theta_0")
139 | label0 = MathTex(r"f(x, \theta_0)", color=BLUE).scale(.8).next_to(self.ax.c2p(self.t0, .75*self.y0), direction=RIGHT)
140 | graph0.set_stroke(opacity = .5)
141 | label0.next_to(self.ax.c2p(self.t0, .75*self.y0), direction=LEFT)
142 | self.add(graph0, label0)
143 |
144 | graph1, area1, area1_label = self.get_piecewise_plot(self.t0 + self.dt, self.y0, .5*self.y0, RED, r"\theta_0 + \Delta\theta")
145 | label1 = MathTex(r"f(x, \theta_0 + \Delta\theta)", color=RED).scale(.8).next_to(self.ax.c2p(self.t0 + self.dt, .75*self.y0), direction=RIGHT)
146 | self.add(graph1, label1, area1, area1_label)
147 |
148 |
149 | class FailureExample_2(FailureExample_1):
150 | def __init__(self):
151 | super().__init__()
152 |
153 | def construct(self):
154 | graph0, _, _ = self.get_piecewise_plot(self.t0, self.y0, .5*self.y0, BLUE, r"\theta_0")
155 | label0 = MathTex(r"f(x, \theta_0)", color=BLUE).scale(.8).next_to(self.ax.c2p(self.t0, self.y0), direction=DL)
156 | graph1, _, _ = self.get_piecewise_plot(self.t0 + self.dt, self.y0, .5*self.y0, RED, r"\theta_0 + \Delta\theta")
157 | label1 = MathTex(r"f(x, \theta_0 + \Delta\theta)", color=RED).scale(.8).next_to(self.ax.c2p(self.t0 + self.dt, .5*self.y0), direction=DR)
158 |
159 | self.add(Polygon(
160 | self.ax.c2p(self.t0 , .5*self.y0),
161 | self.ax.c2p(self.t0 + self.dt, .5*self.y0),
162 | self.ax.c2p(self.t0 + self.dt, self.y0),
163 | self.ax.c2p(self.t0 , self.y0),
164 | stroke_width=0., fill_color=ORANGE, fill_opacity=.3
165 | ))
166 | self.add(graph0, label0, graph1, label1)
167 |
168 | arrow_x = self.t0 + .5*self.dt
169 | arrow_y = .75*self.y0
170 | arrow = Arrow(self.ax.c2p(arrow_x - .15, arrow_y), self.ax.c2p(arrow_x, arrow_y), buff=0., color=ORANGE, max_tip_length_to_length_ratio=.1)
171 | self.add(arrow)
172 |
173 | d_area_label = MathTex(r"I(\theta_0 + \Delta\theta) - I(\theta_0)", color=ORANGE)
174 | d_area_label.scale(.8).next_to(arrow.get_start(), direction=LEFT)
175 | self.add(d_area_label)
176 |
177 | line0 = self.ax.get_vertical_line(self.ax.i2gp(self.t0, graph0[1]), color=BLUE).set_opacity(.5)
178 | line1 = self.ax.get_vertical_line(self.ax.i2gp(self.t0 + self.dt, graph1[1]), color=RED).set_opacity(.5)
179 | self.add(line0, line1)
180 |
181 | dot0 = Dot(self.ax.c2p(self.t0 , 0.), color=BLUE).scale(.8)
182 | dot1 = Dot(self.ax.c2p(self.t0 + self.dt, 0.), color=RED).scale(.8)
183 | self.add(dot0, dot1)
184 |
185 | dot0_label = MathTex(r"x(\theta_0)", color=BLUE).scale(.8).next_to(dot0, direction=.5*UL)
186 | dot1_label = MathTex(r"x(\theta_0 + \Delta\theta)", color=RED).scale(.8).next_to(dot1, direction=.5*UR)
187 | self.add(dot0_label, dot1_label)
188 |
189 | brace = Brace(
190 | Line(self.ax.c2p(self.t0 + self.dt, .5*self.y0), self.ax.c2p(self.t0 + self.dt, self.y0)),
191 | direction=RIGHT, color=BLACK, fill_opacity=.5
192 | )
193 | brace_tex = MathTex(r"\Delta f = \frac{1}{2}", color=BLACK).scale(.8).set_opacity(.5)
194 | brace_tex.next_to(brace, RIGHT)
195 | self.add(brace, brace_tex)
196 |
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/hugo/content/docs/diff-render/basics/diff-int-1d/index.md:
--------------------------------------------------------------------------------
1 | ---
2 | title: "Differentiating 1D Integrals"
3 | weight: 1
4 | mathjax: true
5 | # bookFlatSection: false
6 | # bookToc: true
7 | # bookHidden: false
8 | # bookCollapseSection: false
9 | # bookComments: false
10 | # bookSearchExclude: false
11 | ---
12 |
13 | # Differentiating 1D Integrals
14 |
15 | _by [Shuang Zhao](https://shuangz.com)_
16 |
17 | {{< latex_macros_basic >}}
18 |
19 | In what follows, we discuss the differentiation of a simple Riemann integral `$I(\theta)$` over some 1D interval `$(a, b) \subseteq \real$`:
20 |
21 |
22 | \begin{equation}
23 | \label{eqn:I}
24 | I(\theta) = \int_a^b f(x, \theta) \,\D x.
25 | \end{equation}
26 |
27 |
28 | ## The Incomplete Solution
29 |
30 | The derivative of the integral in Eq. \eqref{eqn:I} with respect to `$\theta$` can _sometimes_ be obtained by exchanging the ordering of differentiation and integration:
31 |
32 |
33 | \begin{equation}
34 | \label{eqn:dI_0}
35 | \frac{\D}{\D\theta} I = \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right)
36 | \stackrel{\Large ?}{=} \int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x.
37 | \end{equation}
38 |
39 |
40 | Precisely, the second equality in Eq. \eqref{eqn:dI_0} requires the integrand $f$ to be **continuous**[^1] throughout the interval `$(a, b)$`.
41 |
42 | [^1]: Unless otherwise stated, we use "continuous" to indicate the `$C^0$` class.
43 |
44 | ### Success Example
45 |
46 | We now provide a toy example where Eq. \eqref{eqn:dI_0} holds.
47 | Let `$f(x, \theta) := x^2 \,\theta$`. Consider the following integral:
48 |
49 |
50 | $$
51 | I = \int_0^1 (x^2 \,\theta) \,\D x.
52 | $$
53 |
54 |
55 | Since `$I = \left[ (x^3 \,\theta)/3 \right]_0^1 = \theta/3$`, we know that
56 |
57 |
58 | $$
59 | \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{\theta}{3} \right)
60 | = {\color{blue}\frac{1}{3}}.
61 | $$
62 |
63 |
64 | We now try calculating the same derivative `$\D I/\D\theta$` using Eq. \eqref{eqn:dI_0}:
65 |
66 |
67 | $$
68 | \frac{\D I}{\D\theta} = \int_0^1 \frac{\D}{\D\theta} (x^2 \,\theta) \,\D x
69 | = \int_0^1 x^2 \,\D x
70 | = \left[ \frac{x^3}{3} \right]_0^1
71 | = {\color{blue}\frac{1}{3}},
72 | $$
73 |
74 |
75 | which matches the manually calculated result above.
76 |
77 | ### Failure Example
78 |
79 | We now show another toy example for which simply exchanging differentiation and integration outlined in Eq. \eqref{eqn:dI_0} fails.
80 | Let
81 |
82 |
83 | \begin{equation}
84 | \label{eqn:f_step}
85 | f(x, \theta) := \begin{cases}
86 | 1, & (x < \theta/2)\\
87 | 1/2. & (x \geq \theta/2)
88 | \end{cases}
89 | \end{equation}
90 |
91 |
92 | Then, for any `$0 < \theta < 2$`, it holds that
93 |
94 |
95 | $$
96 | \begin{split}
97 | I &= \int_0^1 f(x, \theta) \,\D x
98 | = \left( \int_0^{\theta/2} \D x \right) + \left( \int_{\theta/2}^1 \frac{1}{2} \,\D x \right)\\
99 | &= \left[ x \right]_0^{\theta/2} + \left[ \frac{x}{2} \right]_{\theta/2}^1
100 | = \frac{\theta}{2} + \left( \frac{1}{2} - \frac{\theta}{4} \right)
101 | = \frac{1}{2} + \frac{\theta}{4},
102 | \end{split}
103 | $$
104 |
105 |
106 | and
107 |
108 |
109 | \begin{equation}
110 | \label{eqn:f_step_dI_manual}
111 | \frac{\D I}{\D\theta} = \frac{\D}{\D\theta} \left( \frac{1}{2} + \frac{\theta}{4} \right) = {\color{red}\frac{1}{4}}.
112 | \end{equation}
113 |
114 |
115 | However, since the integrand $f$ is piecewise-constant in this example, we have `$\D f/\D\theta \equiv 0$`.
116 | Thus, Eq. \eqref{eqn:dI_0} in this example gives
117 |
118 |
119 | $$
120 | \int_0^1 \frac{\D}{\D\theta} f(x, \theta) \,\D x = \int_0^1 0 \,\D x = {\color{red}0},
121 | $$
122 |
123 |
124 | which does **not** match the manually calculated result in Eq. \eqref{eqn:f_step_dI_manual}.
125 |
126 | ## The General Solution
127 |
128 | ### Examining The Previous Examples
129 |
130 | Before presenting the general expression of the derivative `$\D I/\D\theta$`, we first examine the examples shown above.
131 |
132 | #### The Success Example
133 |
134 | We first examine the [success example](#success-example) with the integrand `$f(x, \theta) = x^2 \,\theta$`.
135 | In the following, we show the graph of `$f(x, \theta)$` for some fixed `$\theta = \theta_0$` in the following:
136 | 
137 | By definition, the integral `$I(\theta_0) := \int_0^1 f(x, \theta) \,\D x$` equals the signed area (marked in light blue) of the region below the graph.
138 | Further, by adding some small `$\Delta\theta > 0$` to `$\theta_0$`, we obtain the graph of `$f(x, \theta_0 + \Delta\theta)$` and the corresponding signed area `$I(\theta_0 + \Delta\theta)$`, both illustrated in red:
139 | 
140 |
141 | We recall that the derivative of $I$ with respect to `$\theta$` is given by the rate at which $I$ changes with `$\theta$`.
142 | To calculate this rate, we examine the difference between `$I(\theta_0 + \Delta\theta)$` and `$I(\theta_0)$`:
143 |
144 |
145 | \begin{equation}
146 | \label{eqn:diffI0_0}
147 | I(\theta_0 + \Delta\theta) - I(\theta_0) = \int_0^1 \left(f(x, \theta_0 + \Delta\theta) - f(x, \theta_0)\right) \,\D x.
148 | \end{equation}
149 |
150 |
151 | Geometrically, this difference equals the (signed) area of the orange region illustrated below:
152 | 
153 |
154 | At each fixed `$0 < x < 1$`, the integrand of Eq. \eqref{eqn:diffI0_0} satisfies that
155 |
156 |
157 | $$
158 | f(x, \theta_0 + \Delta\theta) - f(x, \theta_0) \approx \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta.
159 | $$
160 |
161 |
162 | Base on this relation, we can rewrite the area difference \eqref{eqn:diffI0_0} as:
163 |
164 |
165 | $$
166 | I(\theta_0 + \Delta\theta) - I(\theta_0)
167 | \approx \int_0^1 \left( \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \Delta\theta \right) \D x
168 | = \Delta\theta \int_0^1 \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \D x.
169 | $$
170 |
171 |
172 | In both equations above, the equalities become exact at the limit of `$\Delta\theta \to 0$`.
173 | By dividing both sides by `$\Delta\theta$` and taking the limit of `$\Delta\theta \to 0$`, we have
174 |
175 |
176 | $$
177 | \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0}
178 | := \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta} = \int_0^1 \left[ \frac{\D}{\D\theta} f(x, \theta) \right]_{\theta = \theta_0} \D x,
179 | $$
180 |
181 |
182 | for any `$0 < \theta_0 < 1$`.
183 | This agrees with the incomplete solution expressed in Eq. \eqref{eqn:dI_0}.
184 |
185 | #### The Failure Example
186 |
187 | So what has been the cause for the [failure example](#failure-example)?
188 | To be specific, what has been missing from the incomplete solution \eqref{eqn:dI_0}?
189 |
190 | To understand what has been going on, we again examine the integrand `$f(x, \theta)$` which, for this example, is the piecewise-constant function defined in Eq. \eqref{eqn:f_step}.
191 |
192 | The following are the graphs of `$f(x, \theta)$` for some fixed `$\theta = \theta_0$` and `$\theta = \theta_0 + \Delta\theta$` (for some small `$\Delta\theta > 0$`), respectively:
193 | 
194 | 
195 |
196 | Further, the difference `$I(\theta_0 + \Delta\theta) - I(\theta_0)$` between the signed areas below the two graphs is caused by the rectangle illustrated in orange:
197 | 
198 |
199 | Intuitively, in the [success example](#success-example), the change of signed area is caused by **vertical** shifts of the graph---which is captured by the incomplete solution \eqref{eqn:dI*0}.
200 | On the other hand, in this [failure example](#failure-example), the change of signed area is caused by **horizontal** shifts of the graph \_at jump discontinuities*---which is _missing_ from the incomplete solution!
201 |
202 | We now calculate the signed area of the orange rectangle shown above.
203 | We first observe that the length of the rectangle's vertical edge equals the difference `$\Delta f \equiv 1 - 1/2 = 1/2$` of the integrand `$f(x, \theta)$` across the discontinuity point.
204 |
205 | To calculate the length of the rectangle's horizontal edge, we let `$x(\theta) = \theta/2$` denote the jump discontinuity point of `$f(x, \theta)$` defined in Eq. \eqref{eqn:f_step}.
206 | Then, the (signed) length of the horizontal edge is simply `$x(\theta_0 + \Delta\theta) - x(\theta_0)$`.
207 |
208 | Based on the observations above, we know that
209 |
210 |
211 | $$
212 | I(\theta_0 + \Delta\theta) - I(\theta_0) = \Delta f \,(x(\theta_0 + \Delta\theta) - x(\theta_0)).
213 | $$
214 |
215 |
216 | Dividing both sides of this equation by `$\Delta t$` and taking the limit `$\Delta\theta \to 0$` produce:
217 |
218 |
219 | $$
220 | \begin{split}
221 | \left[ \frac{\D}{\D\theta} I(\theta) \right]_{\theta = \theta_0}
222 | &= \lim_{\Delta\theta \to 0} \frac{I(\theta_0 + \Delta\theta) - I(\theta_0)}{\Delta\theta}\\
223 | &= \Delta f \,\lim_{\Delta\theta \to 0}\frac{x(\theta_0 + \Delta\theta) - x(\theta_0)}{\Delta\theta}
224 | = \Delta f \left[ \frac{\D}{\D\theta} x(\theta) \right]_{\theta = \theta_0}.
225 | \end{split}
226 | $$
227 |
228 |
229 | Therefore, we know that
230 |
231 |
232 | $$
233 | \frac{\D}{\D\theta} I(\theta) = \underbrace{\Delta f}_{=\, 1/2} \; \underbrace{\frac{\D}{\D\theta} x(\theta)}_{=\, 1/2}
234 | = {\color{red}\frac{1}{4}},
235 | $$
236 |
237 |
238 | matching the hand-derived result in Eq. \eqref{eqn:f_step_dI_manual}.
239 |
240 | ### The Full Derivative
241 |
242 | Based on the observations [above](#examining-the-previous-examples), we now present the general derivative of the 1D integral expressed in Eq. \eqref{eqn:I}:
243 |
244 |
245 | \begin{equation}
246 | \label{eqn:dI}
247 | \boxed{
248 | \frac{\D}{\D\theta} \left( \int_a^b f(x, \theta) \,\D x \right)
249 | = \underbrace{\int_a^b \left( \frac{\D}{\D\theta} f(x, \theta) \right) \D x}_{\text{interior}} \,+\,
250 | \underbrace{\sum_i \Delta f(x_i(\theta), \theta) \,\frac{\D}{\D\theta} x_i(\theta)}_{\text{boundary}}\,,
251 | }
252 | \end{equation}
253 |
254 |
255 | which comprises:
256 |
257 | - A **interior** component obtained by exchanging differentiation and integration operations---identical to Eq. \eqref{eqn:dI_0}.
258 |
259 | - A **boundary** component involving a sum over all jump discontinuity points `$\{ x_i(\theta) : i = 1, 2, \ldots \}$`.
260 |
261 | #### Remarks
262 |
263 | Precisely, `$\Delta f(x, \theta)$` in the _boundary_ component is defined as
264 |
265 |
266 | $$
267 | \Delta f(x, \theta) := \lim_{u \uparrow x} f(u, \theta) - \lim_{u \downarrow x} f(u, \theta),
268 | $$
269 |
270 |
271 | where `$\lim_{u \uparrow x}$` and `$\lim_{u \downarrow x}$` denote **one-sided limits** with `$u$` approaching `$x$` from _below_ (i.e., `$u < x$`) and _above_ (i.e., `$u > x$`), respectively.
272 | For any fixed $\theta$, $\Delta f(x, \theta)$ is nonzero (and well-defined) if and only if `$x$` is a jump discontinuity point of `$f(\cdot, \theta)$`.
273 |
274 | Lastly, when the endpoints `$a$` and `$b$` of the integral depend on `$\theta$`, they should be considered as jump discontinuities with `$\Delta f(a, \theta) = -f(a, \theta)$` and `$\Delta f(b, \theta) = f(b, \theta)$`.
275 |
276 | In the [next section]({{}}), we will present a generalization of Eq. \eqref{eqn:dI} that describes derivatives of Lebesgue integrals.
277 |
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12 |
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