### Introduction

RStudio can be used for a lot of the features that Wolfram Mathematica is often used for, and best of all, RStudio and R are open source and free.

Although I had access to Mathematica during the Maths subject in my Data Science course, I prefer to use software that I can continue to use afterwards, so I came up with ways of using it in place of Wolfram Alpha / Mathematica. A lot of it was not very well documented, since it isn’t a common use for it, so hopefully this helps a few others.

NOTE: If you run the code in R-Studio, you can rotate the 3D plots as well

## Differentiation

Differentiation of (3/2)x^2 + 5x + 2 + 70/x

# In R-Studio, you may need to go to Tools -> Install Package and choose Deriv library(Deriv) # Set the function and the derivatives fx <- expression((3/2)*x^2 + 5*x + 2 + 70/x) dfx <- Deriv(fx, "x") ddfx <- Deriv(dfx, "x") # Display output cat(paste("Given f(x) =", fx), "\n", paste("-> f'(x) =", dfx), "\n", paste("-> f''(x) =", ddfx), "\n")

## Given f(x) = (3/2) * x^2 + 5 * x + 2 + 70/x ## -> f'(x) = 3 * x + 5 - 70/x^2 ## -> f''(x) = 140/x^3 + 3

### Plotting a function

In this example, we plot the function **f(x) = 2x + 1 + 50/x**, with x axes from **-5 to 5** and y axes from **-200 to 200**.

# function definition and parameters f <- function(x) 2*x + 1 + 50/x # Create a function to draw x and y axes with dashed lines draw_axes <- function(){ # Draw dashed line for x and y axes abline(h=0, untf=FALSE, col="gray", lwd=1, lty=2) abline(v=0, untf=FALSE, col="gray", lwd=1, lty=2) } # Set the minimum and maximum values for x and y xlim <- c(-5, 5) ylim <- c(-200, 200) # Draw the curve curve(f, main="Your Chart Title", xlim=xlim, ylim=ylim, xname="X Axes Label", ylab="Y Axes Label", col="blue", # color of the line lwd=2, cex.main=2, cex.lab=1.5); # Draw the axes draw_axes()

### Plotting a Relation in 3D

In this example, we plot the relation **f(x,y) = x^3 + 3 xy^2 – 12x + 3y^2** using the Plotly library.

# Use the plotly library to plot our surface library(plotly) # Specify the primary relation as a function called f f <- function (x, y) { return (x^3 + 3*x*y^2 - 12*x + 3*y^2) } # Create values of x and y to plot x <- seq(-4, 4, by=0.25) y <- seq(-3, 3, by=0.25) # calculate z based on the relation for each point of (x, y) z <- outer(x, y, f) # This is really important - transpose z, as plotly expects the data in this format # I spent many hours trying to figure out what was wrong!!! zp <- t(z) # Use Plotly's surface plot feature plot_ly(x = x, y = y, z = zp, type = "surface", colorscale='YlOrRd')

# Note: other available colorscales: # [‘Blackbody’, ‘Bluered’, ‘Blues’, ‘Earth’, ‘Electric’, ‘Greens’, ‘Greys’, ‘Hot’, # ‘Jet’, ‘Picnic’, ‘Portland’, ‘Rainbow’,‘RdBu’,‘Reds’,‘Viridis’,‘YlGnBu’,‘YlOrRd’]

### Creating a Contour Plot of a Relation

Here’s a contour plot of the same function above.

# Set a color scheme for the contour plot cols <- rainbow(60) # NOTE: other available colors are: # terrain.colors(25), cm.colors(25) heat.colors(25, alpha=1, rev=FALSE), # rainbow(25), topo.colors(25) # Draw the contour plot filled.contour(x,y,z, nlevels=25, col=cols, ylab='y', xlab='x', key.title= title('z'))

### Plotting a Relation and Contours in 3D

Using the same relation, this time projecting contour plots in the 3 dimensional plane..

fig <- plot_ly(x=x,y=y,z=zp) %>% add_surface( colorscale = 'YlOrRd', contours = list( z = list( show=TRUE, usecolormap=TRUE, highlightcolor="#ff0000", project=list(z=TRUE), # this projects them onto the plane below/above # control size of lines between contours, and when they start and end size=50, start=0, end=1500, showlines=TRUE ) ) ) fig

### Bounding a relation with another relation

Let’s say that we now only want to see the relation **f(x,y) = x^3 + 3 xy^2 – 12x + 3y^2** plotted where the values exist within the boundary of secondary relation.

In this case the secondary relation is an ellipse specified by **x^2 + 3y^2 = 12**, so we want to bind everything within **x^2 + 3y^2 <= 12**.

# Create function of the first relation that only returns values within the # bounds of the secondary relation fn <- function (x, y) { if ((x^2+3*y^2) <= 12 ) { # The secondary relation out <- (x^3+3*x*y^2-12*x+3*y^2) # Return the value from the primary relation } else { out <- "NULL" } return (out) } # Generate a dataframe called df, with two columns x and y, # containing an array of combinations of x and y that we used earlier df <- merge(data.frame(x=x), data.frame(y=y)) # Create a column holding the result of the relation on x and y vecFn <- Vectorize(fn, vectorize.args = c('x','y')) # Insert this new column into our dataframe df as column z df$z <- vecFn(df$x, df$y) # Now drop all values that are the function fn we created marked it as null # (ie remove items outside of the bounds) df <- subset(df, z != "NULL") # Convert the z column to numeric (since before it had strings of "NULL" in it) df$z <- as.numeric((df$z)) # Finally, plot the dataframe p <- plot_ly(df, x=~df$x, y=~df$y, z=~df$z, type='mesh3d', intensity=~z, colors=colorRamp(rainbow(5))) p

### Projecting the bounded relation onto the primary relation

Lastly, I would like to see how this bounded relation appears on top of the original relation. The result below isn’t perfect, but it gives you an idea.

# plot the two together using the subplot primary_surface <- plot_ly(x = x, y = y, z = zp, type = "surface", colorscale='YlOrRd') bounded_surface <- plot_ly(df, x=~df$x, y=~df$y, z=~df$z, type='mesh3d', intensity=~z, colors=colorRamp(rainbow(5)) ) subplot(primary_surface, bounded_surface)