# Problem 1

library(ggplot2)
diamonds
names(diamonds)
# [1] "carat" "cut" "color" "clarity" "depth"
# [6] "table" "price" "x"    "y"    "z"

# 1
summary(diamonds)
# carat             cut    color    clarity       depth       table
# Min. :0.2000 Fair : 1610 D: 6775 SI1 :13065 Min. :43.00 Min. :43.00
# 1st Qu.:0.4000 Good : 4906 E: 9797 VS2 :12258 1st Qu.:61.00 1st Qu.:56.00
# Median :0.7000 Very Good:12082 F: 9542 SI2 : 9194 Median :61.80 Median :57.00
# Mean :0.7979 Premium :13791 G:11292 VS1 : 8171 Mean :61.75 Mean :57.46
# 3rd Qu.:1.0400 Ideal :21551 H: 8304 VVS2 : 5066 3rd Qu.:62.50 3rd Qu.:59.00
# Max. :5.0100                   I: 5422 VVS1 : 3655 Max. :79.00 Max. :95.00
#                                  J: 2808 (Other): 2531
# price          x             y             z
# Min. : 326 Min. : 0.000 Min. : 0.000 Min. : 0.000
# 1st Qu.: 950 1st Qu.: 4.710 1st Qu.: 4.720 1st Qu.: 2.910
# Median : 2401 Median : 5.700 Median : 5.710 Median : 3.530
# Mean : 3933 Mean : 5.731 Mean : 5.735 Mean : 3.539
# 3rd Qu.: 5324 3rd Qu.: 6.540 3rd Qu.: 6.540 3rd Qu.: 4.040
# Max. :18823 Max. :10.740 Max. :58.900 Max. :31.800

cor(diamonds[c("carat", "depth", "table", "price", "x", "y", "z")])
#       carat    depth    table    price       x       y       z
# carat 1.00000000 0.02822431 0.1816175 0.9215913 0.97509423 0.95172220 0.95338738
# depth 0.02822431 1.00000000 -0.2957785 -0.0106474 -0.02528925 -0.02934067 0.09492388
# table 0.18161755 -0.29577852 1.0000000 0.1271339 0.19534428 0.18376015 0.15092869
# price 0.92159130 -0.01064740 0.1271339 1.0000000 0.88443516 0.86542090 0.86124944
# x 0.97509423 -0.02528925 0.1953443 0.8844352 1.00000000 0.97470148 0.97077180
# y 0.95172220 -0.02934067 0.1837601 0.8654209 0.97470148 1.00000000 0.95200572
# z 0.95338738 0.09492388 0.1509287 0.8612494 0.97077180 0.95200572 1.00000000

ggplot(diamonds, aes(x=carat, y=price, col=color)) + geom_point()
# color: diamond colour, from D (best) to J (worst)

Find the gradient field F = VQ for the potential function p(x,y) =...