diff --git a/book/numerical_methods/1-revision-of-concepts.ipynb b/book/numerical_methods/1-revision-of-concepts.ipynb
index 5a24cb565cde8603ab6cda50e061713306f42e23..5f4d37ce1bd03a44d7546f3797dbb064963a818f 100644
--- a/book/numerical_methods/1-revision-of-concepts.ipynb
+++ b/book/numerical_methods/1-revision-of-concepts.ipynb
@@ -23,7 +23,7 @@
"```{note}\n",
"**Important things to retain from this block:**\n",
"* Identify characteristics of differential equations \n",
- "* Understanding how numerical and analytical solutions might differ as we get to more complex problems which need simplification\n",
+ "* Understand how numerical and analytical solutions might differ as we get to more complex problems which need simplification\n",
"* Remember that analytical solutions are not always possible\n",
"```"
]
@@ -32,7 +32,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Differential equations can be classified in Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). **ODEs** have derivatives with respect to a **single** independent variable (either time **or** space), for example \n",
+ "Differential equations can be classified as Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). **ODEs** have derivatives with respect to a **single** independent variable (either time **or** space), for example: \n",
"\n",
"$$ \n",
"\\frac{d x(t)}{d t} = \\cos t\n",
@@ -47,7 +47,7 @@
"$$\n",
"describes the propagation of the concentration $c$ in **time $t$** along **dimension $x$**.\n",
"\n",
- "The classification can be done more precisely, by its order and linearity (linear or non-linear). The order refers to the highest derivative while non-linear equations are those in which the dependent variable or its derivative(s) are non linear (exponential, sine or with a power different than 1). See the following examples:\n",
+ "The classification can be done more precisely by the equation's order and linearity (linear or non-linear). The order refers to the highest derivative while non-linear equations are those in which the dependent variable or its derivative(s) are non linear (exponential, sine or with a power different than 1). See the following examples:\n",
"\n",
"$$\n",
"\\frac{d^3y}{dx^3} - x\\frac{d^2y}{dx^2} + y = 0 \\qquad \\text{third-order linear ODE}\n",
@@ -78,7 +78,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Equations can be solved in two ways: anaylitically and numerically. The analytical solution is **exact** while a numerical solution requires computational methods to **approximate** the solution. So why would we use anything other than analytical solutions? Well, analytical solutions are difficult/impossible to find for complex equations, specially when the problem involves a complex geometry. This complexity will be treated later in the book. For now, let's illustriate analytical and numerical solutions considering a simple problem.\n",
+ "Equations can be solved in two ways: anaylitically and numerically. The analytical solution is **exact** while a numerical solution requires computational methods to **approximate** the solution. So, why would we use anything other than analytical solutions? Well, analytical solutions are difficult/impossible to find for complex equations, specially when the problem involves a complex geometry. This complexity will be treated later in the book. For now, let's illustriate analytical and numerical solutions considering a simple problem.\n",
"\n",
":::{card}\n",
"> Find the value $x$ for $f(x)=3x-2$ when $f(x) = 0$ in the interval $[0,1]$.\n",
diff --git a/book/numerical_methods/2-derivative.ipynb b/book/numerical_methods/2-derivative.ipynb
index b2ede310db4757283373a0154ba8838a7035db8f..d42e11212d5e992137617adc02447351274ce5d6 100644
--- a/book/numerical_methods/2-derivative.ipynb
+++ b/book/numerical_methods/2-derivative.ipynb
@@ -49,7 +49,7 @@
"\n",
"\n",
"\n",
- "**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at the figure to the right. The first derivative evaluated at $x_0$ is illustrated as a simple slope. You can also see that the rate of change at $x_0$ is larger than at $x_1$. "
+ "**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at the figure above. The first derivative evaluated at $x_0$ is illustrated as a simple slope, as is the first derivative evaluated at $x_1$ . You can also see that the rate of change at $x_0$ is larger than at $x_1$. "
]
},
{
@@ -59,7 +59,7 @@
"## Numerical Freedom to compute derivatives\n",
"\n",
"\n",
- "In the Figure above, the derivative approximation was illustrated arbitarly using two points: the one at which the derivate was evaluated and another point in front of it. However, there are more possibilities. Instead of using points at $x_{-1,0,1}$ a more general notation is used: $x_{i-1,i,i+1}$. The simplest ways to approximate the derivative evaluated at the pont $x_i$ use two points: \n",
+ "In the Figure above, the derivative approximation was illustrated arbitarly using two points: the one at which the derivate was evaluated and another point in front of it. However, there are more possibilities. Instead of using absolute points at $x_{-1,0,1}$ a more general notation is used: $x_{i-1,i,i+1}$. The simplest ways to approximate the derivative evaluated point $x_i$ is to use two points: \n",
"\n",
"$$\n",
"\\text{forward: }\\hspace{3mm} \\frac{df}{dx}\\bigg\\rvert_{x_i}\\approx\\frac{f(x_{i+1})-f(x_{i})}{x_{i+1}-x_i} \\hspace{5mm} \\text{backward: } \\hspace{3mm} \\frac{df}{dx}\\bigg\\rvert_{x_i}\\approx\\frac{f(x_{i})-f(x_{i-1})}{x_{i}-x_{i-1}} \\hspace{5mm} \\text{central } \\hspace{3mm} \\frac{df}{dx}\\bigg\\rvert_{x_i}\\approx\\frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}\n",
diff --git a/book/numerical_methods/3-taylor-series-expansion.ipynb b/book/numerical_methods/3-taylor-series-expansion.ipynb
index daf901f3f98ab79abb3114698814b02280db781c..490a4905cc8cf55aa3846aec326b9e377be6fa83 100644
--- a/book/numerical_methods/3-taylor-series-expansion.ipynb
+++ b/book/numerical_methods/3-taylor-series-expansion.ipynb
@@ -13,7 +13,7 @@
"source": [
"```{note}\n",
"**Important things to retain from this block:**\n",
- "* Understand how to compute Taylor series expansion of a given function around a given point and its limitations\n",
+ "* Understand how to compute Taylor series expansion (TSE) of a given function around a given point and its limitations\n",
"* Understand how numerical derivatives are derived from TSE\n",
"* Understand that the magnitude error depends on the expression used\n",
"**Things you do not need to know:**\n",
@@ -33,6 +33,7 @@
"This series is **exact** as long as we include infinite terms. We, however, are limited to a **truncated** expression: an **approximation** of the real function.\n",
"\n",
"Using only 3 terms and defining $\\Delta x=x-x_i$, the TSE can be rewritten as \n",
+ "\n",
"$$f(x_i+\\Delta x) = f(x_i) + \\Delta x f'(x_i)+\\frac{\\Delta x^2}{2!}f''(x_i)+ \\frac{\\Delta x^3}{3!} f'''(x_i)+ \\mathcal{O}(\\Delta x)^4.$$\n",
"\n",
"Here $\\Delta x$ is the distance between the point we \"know\" and the desired point. $\\mathcal{O}(\\Delta x)^4$ means that we do not take into account the terms associated to $\\Delta x^4$ and therefore **that is the truncation error order**. From here we can also conclude that **the larger the step $\\Delta x$, the larger the error**!\n",
@@ -101,7 +102,7 @@
"\n",
"How good is this polynomial? How does the result varies on the number of terms used?\n",
"\n",
- " Press `rocket` -->`Live Code` to interact with the figure\n",
+ " Press `rocket` -->`Live Code` to interact with the figure. You should then click 'run all' in the cell below.\n",
"\n",
":::"
]
@@ -174,7 +175,7 @@
"Relevant conclusions:\n",
"- The 1st order, which depends only on the first derivative evaluation, is a straight line. \n",
"- The more terms used (larger order) the smaller the error.\n",
- "- The farther from the starting point (e.g., in the plots $x_i=0$), the larger the error. "
+ "- The further from the starting point (e.g., in the plots $x_i=0$), the larger the error. "
]
},
{
@@ -218,7 +219,7 @@
"f'(x_i)=\\frac{f(x_i+\\Delta x)-f(x_i-\\Delta x)}{2\\Delta x}+ \\mathcal{O}(\\Delta x^2).\n",
"$$\n",
"\n",
- "The second derivative terms cancel each other, therefore **the order error is the step size squared!** From here, it is obvious that the central difference is more accurate. You can notice it as well intuitively in the figure of the previous chapter. \n"
+ "The second derivative terms cancel each other out, therefore **the order error is the step size squared!** From here, it is obvious that the central difference is more accurate. You can notice it as well intuitively in the figure of the previous chapter. \n"
]
},
{
@@ -321,7 +322,7 @@
"source": [
"## TSE to define second derivatives\n",
"\n",
- "There are equations that require second derivatives. The diffusion equation is one of those used in every field of knowledge. The 1-D diffusion equation reads \n",
+ "There are equations that require second derivatives. One example is the diffusion equation. The 1-D diffusion equation reads: \n",
"\n",
"$$\n",
"\\frac{\\partial f}{\\partial t}=v\\frac{\\partial^2 f}{\\partial x^2} \\text{ where } v \\text{ is the diffusion coefficient.}\n",
@@ -329,7 +330,7 @@
"\n",
"For the moment we will use TSE to find **only** a numerical expression of the second derivative $\\frac{\\partial^2 f}{\\partial x^2}$.\n",
"\n",
- "The procedure is simple but cumbersome. The general idea is to isolate the second derivative in the TSE without being a dependency on other derivatives. Below you can find more details about the algebraic manipulation (if you are curious) but you do not need to know it. Here is the result: \n",
+ "The procedure is simple but cumbersome. The general idea is to isolate the second derivative in the TSE without there being a dependency on other derivatives. Below you can find more details about the algebraic manipulation (if you are curious) but you do not need to know it. Here is the result: \n",
"\n",
"$$\n",
"f''(x_i)=\\frac{f(x_i+2\\Delta x)-2f(x_i+\\Delta x)+f(x_i)}{\\Delta x^2}+ \\mathcal{O}(\\Delta x).\n",
@@ -351,7 +352,7 @@
"f(x_i+2\\Delta x) = f(x_i) + 2\\Delta x f'(x_i)+\\frac{(2\\Delta x)^2}{2!}f''(x_i)+ \\mathcal{O}(\\Delta x)^3.\n",
"$$\n",
"\n",
- "Now multiply by two the TSE for a point one step farher from $x_i$: \n",
+ "Now multiply the TSE by two for a point one step farher from $x_i$: \n",
"\n",
"$$\n",
"2f(x_i+\\Delta x) = 2f(x_i) + 2\\Delta x f'(x_i)+\\frac{2\\Delta x^2}{2!}f''(x_i) + \\mathcal{O}(\\Delta x)^3.\n",
diff --git a/book/numerical_methods/4-numerical-integration.ipynb b/book/numerical_methods/4-numerical-integration.ipynb
index 5eb47e1ce38cf18f9922a4d43116f55004717ae6..327bdfe99201980f57ce7d6734d66663d917b653 100644
--- a/book/numerical_methods/4-numerical-integration.ipynb
+++ b/book/numerical_methods/4-numerical-integration.ipynb
@@ -14,8 +14,8 @@
"NOTE\n",
"```{note}\n",
"**Important things to retain from this block:**\n",
- "* Knowing how to apply different discrete integration techniques\n",
- "* Having an idea on how the truncation errors change from technique to technique \n",
+ "* Know how to apply different discrete integration techniques\n",
+ "* Have an idea on how the truncation errors change from technique to technique \n",
"```"
]
},
@@ -41,7 +41,7 @@
"\n",
"\n",
"\n",
- "In a 1 dimension situation, the integral\n",
+ "In a 1 dimension situation, the integral:\n",
"\n",
"$$ \n",
"I=\\int_a^b f(x)dx\n",
@@ -80,7 +80,7 @@
"\n",
"Using the lowest polynomial order leads to the Riemann integral also known as the rectangle rule. Basically, the function is divided in segments with a known $\\Delta x$ and the area of each segment is represented by a rectangle. Its height, related to the discretized points of the function, can be defined by the left or by the right. \n",
"\n",
- "Therefore, we get the left Riemann integration formula given by\n",
+ "Therefore, we get the left Riemann integration formula given by:\n",
"\n",
"$$\n",
"\\int_{a}^{b} f(x) \\, dx \\approx \\sum_{i=0}^{n-1} f(x_i)\\Delta x, \\hspace{5mm} \\Delta x = \\frac{b - a}{n}\n",
diff --git a/book/numerical_methods/overview.md b/book/numerical_methods/overview.md
index 4369b0300a63944b3fe70c7b54a975a45e73e19b..ee9aff7e1bfec42fa5396514b40e98883102ed4b 100644
--- a/book/numerical_methods/overview.md
+++ b/book/numerical_methods/overview.md
@@ -1,7 +1,7 @@
# Bridging the Gap in Civil Engineering & Geoscience
-Civil engineering and geosciences now modernised to augment experimental observations with precise numerical techniques, for the analysis and design complex structures. In this regard, the Finite Difference Method (FDM) is a fundamental and indispensable numerical approach, which plays a pivotal role in solving differential equations that govern physical phenomena.
+Civil Engineering and Geosciences are now modernised to augment experimental observations with precise numerical techniques, for the analysis and design complex structures. In this regard, the Finite Difference Method (FDM) is a fundamental and indispensable numerical approach, which plays a pivotal role in solving differential equations that govern physical phenomena.
These phenomena include and are not limited to heat flow through media, the distribution of stresses in solid structures, fluid flow through porous media, and pollutant dispersion in the air and water. Solving the differential equations that govern these phenomena is often infeasible for the complex geometries and boundary conditions that are encountered in engineering.
-This is where these numerical methods and FDMs come in indispensable as it systematically discretizes the expressions, transforming them into algebraic equations that can be solved numerically on a computer. This approach provides engineers with the flexibility to model and analyze a wide range of scenarios, optimize designs, ensure safety, reduce costs, and predict failures too.
\ No newline at end of file
+This is where these numerical methods and FDMs become indispensable as it systematically discretizes the expressions, transforming them into algebraic equations that can be solved numerically on a computer. This approach provides engineers with the flexibility to model and analyze a wide range of scenarios, optimize designs, ensure safety, reduce costs, and predict failures.
\ No newline at end of file
diff --git a/book/numerical_methods/taylor-series-exercise.ipynb b/book/numerical_methods/taylor-series-exercise.ipynb
index cb67dca506da70d19988519116f525619b15bcc3..97888ab0641a12ab20ecd8c5bea33096bc72a57a 100644
--- a/book/numerical_methods/taylor-series-exercise.ipynb
+++ b/book/numerical_methods/taylor-series-exercise.ipynb
@@ -6,7 +6,7 @@
"source": [
"# Exercises on Taylor expansion\n",
"\n",
- "This page shows some exercises on calculation Taylor expansion. If you reload this page, you'll get new values.\n",
+ "This page shows some exercises on calculating Taylor expansions. If you reload this page, you'll get new values.\n",
"\n",
" Click `rocket` -->`Live Code` to start practising.\n"
]
@@ -71,7 +71,7 @@
"\n",
"## Exercise 1\n",
"\n",
- "Calculate the taylor series expension of:\n"
+ "Calculate the taylor series expansion of:\n"
]
},
{