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The 20 Most Important Mathematical Equations | ||
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A journey through the most elegant and influential formulas in mathematics | ||
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||
1. Euler's Identity | ||
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||
\[ e^{i\pi} + 1 = 0 \] | ||
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||
Connects five fundamental constants \((e, i, \pi, 1, 0)\), revealing the profound relationship between exponential functions and trigonometry. | ||
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||
2. Pythagorean Theorem | ||
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||
\[ a^2 + b^2 = c^2 \] | ||
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||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
3. The Fundamental Theorem of Calculus | ||
|
||
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] | ||
|
||
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | ||
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||
4. Maxwell's Equations | ||
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||
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \] | ||
\[ \nabla \cdot \mathbf{B} = 0 \] | ||
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \] | ||
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] | ||
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||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
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The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
1. Euler's Identity | ||
\[ e^{i\pi} + 1 = 0 \] | ||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
2. Pythagorean Theorem | ||
\[ a^2 + b^2 = c^2 \] | ||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
3. The Fundamental Theorem of Calculus | ||
\[ \int_{a}^{b} f(x)\,dx = F(b) - F(a) \] | ||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. | ||
|
||
4. Maxwell's Equations | ||
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \] | ||
\[ \nabla \cdot \mathbf{B} = 0 \] | ||
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \] | ||
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] | ||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
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Made with | ||
The 20 Most Important Mathematical Equations | ||
A journey through the most elegant and influential formulas in mathematics | ||
1. Euler’s Identity | ||
2. Pythagorean Theorem | ||
e | ||
iττ + 1 = 0 | ||
a2 + b2 = c2 | ||
3. The Fundamental Theorem of Calculus | ||
4. Maxwell’s Equations | ||
Establishes that differentiation and integration are inverse | ||
operations. If F is an antiderivative of f, the definite integral | ||
equals F(b) - F(a). Revolutionized mathematical problem- | ||
solving. | ||
In right triangles, the hypotenuse squared equals the sum of | ||
the squares of the other sides. Cornerstone of geometry with | ||
applications in navigation and architecture. | ||
∇· E = ∂ | ||
ε0 | ||
∇· B = 0 | ||
∇× E = −∂B | ||
∂t | ||
∇× B = µ0J + µ0ε0 ∂E | ||
∂t | ||
Unified electricity and magnetism as manifestations of the | ||
same force. Describes electromagnetic field behavior, predicting | ||
waves traveling at light speed. Enabled technologies from radio | ||
to smartphones. | ||
1. Euler’s Identity | ||
2. Pythagorean Theorem | ||
e | ||
iττ | ||
+ 1 = 0 | ||
a2 + b2 = c2 | ||
3. The Fundamental THEorem of Calculus | ||
4. Maxwell’s Equations |
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## **The 20 Most Important Mathematical Equations** | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
**1. Euler's Identity** Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* + <sup>2</sup> *b* = <sup>2</sup> *c* 2 | ||
|
||
## **3. The Fundamental Theorem of Calculus** | ||
|
||
## **4. Maxwell's Equations** | ||
|
||
$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$ | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problemsolving. | ||
|
||
∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E** | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
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## **The 20 Most Important Mathematical Equations** | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
**1. Euler's Identity** Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* + <sup>2</sup> *b* = <sup>2</sup> *c* 2 | ||
|
||
## **3. The Fundamental Theorem of Calculus** | ||
|
||
## **4. Maxwell's Equations** | ||
|
||
$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$ | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problemsolving. | ||
|
||
∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E** | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
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Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,17 @@ | ||
## **The 20 Most Important Mathematical Equations** | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
**1. Euler's Identity** Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* + <sup>2</sup> *b* = <sup>2</sup> *c* 2 | ||
|
||
## **3. The Fundamental Theorem of Calculus** | ||
|
||
## **4. Maxwell's Equations** | ||
|
||
$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$ | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problemsolving. | ||
|
||
∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E** | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
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Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,17 @@ | ||
## **The 20 Most Important Mathematical Equations** | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
**1. Euler's Identity** Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* + <sup>2</sup> *b* = <sup>2</sup> *c* 2 | ||
|
||
## **3. The Fundamental Theorem of Calculus** | ||
|
||
## **4. Maxwell's Equations** | ||
|
||
$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$ | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problemsolving. | ||
|
||
∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E** | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
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Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,17 @@ | ||
## **The 20 Most Important Mathematical Equations** | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
**1. Euler's Identity** Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. **2. Pythagorean Theorem** In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. *e* + *iπ* 1 = 0 *a* + <sup>2</sup> *b* = <sup>2</sup> *c* 2 | ||
|
||
## **3. The Fundamental Theorem of Calculus** | ||
|
||
## **4. Maxwell's Equations** | ||
|
||
$$\int\_{a}^{b} f(\mathbf{x}) \, d\mathbf{x} = F(b) - F(a)$$ | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problemsolving. | ||
|
||
∇ ⋅ **E** = *ε*0 *ρ* ∇ ⋅ **B** = 0 ∇ × **E** = − ∂*t* ∂**B** ∇ × **B** = *μ*0**J** + *μ*0*ε*<sup>0</sup> ∂*t* ∂**E** | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
19 changes: 19 additions & 0 deletions
19
olmocr/bench/sample_data/olmocr_base_temp0_1/mathfuncs_1.md
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# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
| 1. Euler's Identity | 2. Pythagorean Theorem | | ||
|--------------------|------------------------| | ||
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) | | ||
|
||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations | | ||
|----------------------------------------|------------------------| | ||
| \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \) | \( \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \) | | ||
|
||
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
19 changes: 19 additions & 0 deletions
19
olmocr/bench/sample_data/olmocr_base_temp0_1/mathfuncs_2.md
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# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
| 1. Euler's Identity | 2. Pythagorean Theorem | | ||
|---------------------|------------------------| | ||
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) | | ||
|
||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations | | ||
|----------------------------------------|------------------------| | ||
| \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] | \[ \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \] | | ||
|
||
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
19 changes: 19 additions & 0 deletions
19
olmocr/bench/sample_data/olmocr_base_temp0_1/mathfuncs_3.md
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@@ -0,0 +1,19 @@ | ||
# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
| 1. Euler's Identity | 2. Pythagorean Theorem | | ||
|--------------------|------------------------| | ||
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) | | ||
|
||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations | | ||
|----------------------------------------|------------------------| | ||
| \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \) | \( \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \) | | ||
|
||
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
19 changes: 19 additions & 0 deletions
19
olmocr/bench/sample_data/olmocr_base_temp0_1/mathfuncs_4.md
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---|---|---|
@@ -0,0 +1,19 @@ | ||
# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
| 1. Euler's Identity | 2. Pythagorean Theorem | | ||
|--------------------|------------------------| | ||
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) | | ||
|
||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations | | ||
|----------------------------------------|------------------------| | ||
| \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \) | \( \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \) | | ||
|
||
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
19 changes: 19 additions & 0 deletions
19
olmocr/bench/sample_data/olmocr_base_temp0_1/mathfuncs_5.md
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---|---|---|
@@ -0,0 +1,19 @@ | ||
# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
| 1. Euler's Identity | 2. Pythagorean Theorem | | ||
|--------------------|------------------------| | ||
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) | | ||
|
||
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
| 3. The Fundamental Theorem of Calculus | 4. Maxwell's Equations | | ||
|----------------------------------------|------------------------| | ||
| \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \) | \( \nabla \cdot \mathbf{E} = \frac{Q}{\varepsilon_0} \) | | ||
|
||
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
42 changes: 42 additions & 0 deletions
42
olmocr/bench/sample_data/olmocr_base_temp0_8/mathfuncs_1.md
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# The 20 Most Important Mathematical Equations | ||
|
||
A journey through the most elegant and influential formulas in mathematics | ||
|
||
--- | ||
|
||
**1. Euler's Identity** | ||
|
||
\[ e^{i\pi} + 1 = 0 \] | ||
|
||
Connects five fundamental constants (\(e, i, \pi, 1, 0\)), revealing the profound relationship between exponential functions and trigonometry. | ||
|
||
--- | ||
|
||
**2. Pythagorean Theorem** | ||
|
||
\[ a^2 + b^2 = c^2 \] | ||
|
||
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. | ||
|
||
--- | ||
|
||
**3. The Fundamental Theorem of Calculus** | ||
|
||
\[ \int_a^b f(x) \, dx = F(b) - F(a) \] | ||
|
||
Establishes that differentiation and integration are inverse operations. If \(F\) is an antiderivative of \(f\), the definite integral equals \(F(b) - F(a)\). Revolutionized mathematical problem-solving. | ||
|
||
--- | ||
|
||
**4. Maxwell's Equations** | ||
|
||
\[ | ||
\begin{align*} | ||
\nabla \cdot \mathbf{E} &= \frac{Q}{\varepsilon_0} \\ | ||
\nabla \cdot \mathbf{B} &= 0 \\ | ||
\n\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ | ||
\n\nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} | ||
\end{align*} | ||
\] | ||
|
||
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
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