rssn 0.2.9

A comprehensive scientific computing library for Rust, aiming for feature parity with NumPy and SymPy.
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214

//! # Symbolic Vector Calculus Operations
//!
//! This module provides functions for performing symbolic vector calculus operations,
//! including line integrals (scalar and vector fields), surface integrals (flux),
//! and volume integrals. It defines structures for `ParametricCurve`, `ParametricSurface`,
//! and `Volume` to represent the domains of integration.

use crate::symbolic::calculus::definite_integrate;
use crate::symbolic::calculus::substitute;
use crate::symbolic::core::Expr;
use crate::symbolic::simplify_dag::simplify;
use crate::symbolic::vector::Vector;
use crate::symbolic::vector::partial_derivative_vector;

/// Represents a parameterized curve C given by r(t).
#[derive(Debug, Clone, serde::Serialize, serde::Deserialize)]
pub struct ParametricCurve {
    /// The vector expression for the curve, e.g., [cos(t), sin(t), t].
    pub r: Vector,
    /// The name of the parameter, e.g., "t".
    pub t_var: String,
    /// The integration bounds for the parameter, e.g., (0, 2*pi).
    pub t_bounds: (Expr, Expr),
}

/// Represents a parameterized surface S given by r(u, v).
#[derive(Debug, Clone, serde::Serialize, serde::Deserialize)]
pub struct ParametricSurface {
    /// The vector expression for the surface, e.g., [u*cos(v), u*sin(v), v].
    pub r: Vector,
    /// The name of the first parameter, e.g., "u".
    pub u_var: String,
    /// The integration bounds for the first parameter, e.g., (0, 1).
    pub u_bounds: (Expr, Expr),
    /// The name of the second parameter, e.g., "v".
    pub v_var: String,
    /// The integration bounds for the second parameter, e.g., (0, 2*pi).
    pub v_bounds: (Expr, Expr),
}

/// Represents a volume V for triple integration.
/// Defines the integration order as dz dy dx.
#[derive(Debug, Clone, serde::Serialize, serde::Deserialize)]
pub struct Volume {
    /// The bounds for the innermost integral (dz). Can be expressions in terms of x and y.
    pub z_bounds: (Expr, Expr),
    /// The bounds for the middle integral (dy). Can be expressions in terms of x.
    pub y_bounds: (Expr, Expr),
    /// The bounds for the outermost integral (dx). Must be constants.
    pub x_bounds: (Expr, Expr),
    /// The variable names for (x, y, z).
    pub vars: (String, String, String),
}

/// Computes the line integral of a scalar field `f` along a parameterized curve C.
///
/// The integral is `∫_C f ds = ∫_a^b f(r(t)) ||r'(t)|| dt`.
///
/// # Arguments
/// * `scalar_field` - The scalar field `f` as an `Expr`.
/// * `curve` - The `ParametricCurve` representing the path of integration.
///
/// # Returns
/// An `Expr` representing the symbolic line integral.
#[must_use]
pub fn line_integral_scalar(
    scalar_field: &Expr,
    curve: &ParametricCurve,
) -> Expr {
    let r_prime = partial_derivative_vector(&curve.r, &curve.t_var);

    let r_prime_magnitude = r_prime.magnitude();

    let sub = |expr: &Expr| {
        let e1 = substitute(expr, "x", &curve.r.x);

        let e2 = substitute(&e1, "y", &curve.r.y);

        substitute(&e2, "z", &curve.r.z)
    };

    let field_on_curve = sub(scalar_field);

    let integrand = simplify(&Expr::new_mul(field_on_curve, r_prime_magnitude));

    let integral = definite_integrate(
        &integrand,
        &curve.t_var,
        &curve.t_bounds.0,
        &curve.t_bounds.1,
    );

    simplify(&integral)
}

/// Computes the line integral of a vector field `F` along a parameterized curve C (work).
///
/// The integral is `∫_C F · dr = ∫_a^b F(r(t)) · r'(t) dt`.
///
/// # Arguments
/// * `vector_field` - The vector field `F` as a `Vector`.
/// * `curve` - The `ParametricCurve` representing the path of integration.
///
/// # Returns
/// An `Expr` representing the symbolic line integral.
#[must_use]
pub fn line_integral_vector(
    vector_field: &Vector,
    curve: &ParametricCurve,
) -> Expr {
    let r_prime = partial_derivative_vector(&curve.r, &curve.t_var);

    let sub = |expr: &Expr| {
        let e1 = substitute(expr, "x", &curve.r.x);

        let e2 = substitute(&e1, "y", &curve.r.y);

        substitute(&e2, "z", &curve.r.z)
    };

    let field_on_curve = Vector::new(
        sub(&vector_field.x),
        sub(&vector_field.y),
        sub(&vector_field.z),
    );

    let integrand = field_on_curve.dot(&r_prime);

    let integral = definite_integrate(
        &integrand,
        &curve.t_var,
        &curve.t_bounds.0,
        &curve.t_bounds.1,
    );

    simplify(&integral)
}

/// Computes the surface integral (flux) of a vector field F over a parameterized surface S.
///
/// The integral is `∫∫_S (F · dS) = ∫∫_D F(r(u,v)) · (r_u × r_v) du dv`.
///
/// # Arguments
/// * `field` - The vector field `F` as a `Vector`.
/// * `surface` - The `ParametricSurface` representing the surface of integration.
///
/// # Returns
/// An `Expr` representing the symbolic surface integral.
#[must_use]
pub fn surface_integral(
    field: &Vector,
    surface: &ParametricSurface,
) -> Expr {
    let r_u = partial_derivative_vector(&surface.r, &surface.u_var);

    let r_v = partial_derivative_vector(&surface.r, &surface.v_var);

    let normal_vector = r_u.cross(&r_v);

    let sub = |expr: &Expr| {
        let e1 = substitute(expr, "x", &surface.r.x);

        let e2 = substitute(&e1, "y", &surface.r.y);

        substitute(&e2, "z", &surface.r.z)
    };

    let field_on_surface = Vector::new(sub(&field.x), sub(&field.y), sub(&field.z));

    let integrand = field_on_surface.dot(&normal_vector);

    let inner_integral = definite_integrate(
        &integrand,
        &surface.u_var,
        &surface.u_bounds.0,
        &surface.u_bounds.1,
    );

    let outer_integral = definite_integrate(
        &inner_integral,
        &surface.v_var,
        &surface.v_bounds.0,
        &surface.v_bounds.1,
    );

    simplify(&outer_integral)
}

/// Computes the volume integral of a scalar field `f` over a defined volume V.
///
/// The integral is `∫∫∫_V f dV = ∫_x1^x2 ∫_y1^y2 ∫_z1^z2 f(x,y,z) dz dy dx`.
///
/// # Arguments
/// * `scalar_field` - The scalar field `f` as an `Expr`.
/// * `volume` - The `Volume` structure defining the integration domain and order.
///
/// # Returns
/// An `Expr` representing the symbolic volume integral.
#[must_use]
pub fn volume_integral(
    scalar_field: &Expr,
    volume: &Volume,
) -> Expr {
    let (x_var, y_var, z_var) = (&volume.vars.0, &volume.vars.1, &volume.vars.2);

    let integral_z =
        definite_integrate(scalar_field, z_var, &volume.z_bounds.0, &volume.z_bounds.1);

    let integral_y = definite_integrate(&integral_z, y_var, &volume.y_bounds.0, &volume.y_bounds.1);

    let integral_x = definite_integrate(&integral_y, x_var, &volume.x_bounds.0, &volume.x_bounds.1);

    simplify(&integral_x)
}