Thread by @pgeerkens

@GWOMaths @MathsTim Don't be intimidated!

This problem is just a straightforward calculation exercise requiring a few definitions.

Those definitions use calculus, but I will give some simple rules to follow to get the calculus right.

This thread may take a few minutes to complete;
bear with me.

@GWOMaths @MathsTim The total derivative (with respect x)
of a polynomial function in one variable x
is the sum of the total derivatives of each term
using this rule

The total derivative of a power function in x is evaluated as:
derivative_wrt_x( c.x^n ) = n.c.x^(n-1)

Evaluate each term & add.

@GWOMaths @MathsTim Note that the total derivative of all constant terms is zero.

This results in another function of x,
of order one less than the original.

Evaluated at each point x,
it represents the slope of the tangent line
to the original function
at that point x.

@GWOMaths @MathsTim The partial derivative
with respect to x
of a function f in the three variables (x,y,z)
is calculated as the total derivative wrt x
assuming that y and z are regarded as constants.

So terms wholly in y and z go to 0.

@GWOMaths @MathsTim We have been given a column vector in (x,y,z) that maps each point in R^3 to another point in R^3.

Each component of this column vector
is a polynomial function in (x,y,z),
so we can calculate all the partial derivatives
wrt x and y and z
by the rules above.

@GWOMaths @MathsTim The Jacobean Matrix of our column vector V
is a 3x3 matrix of the partial derivatives
of each component of V by each of x and y and z:

| Px(V1) Py(V1) Pz(V1) |
| Px(V2) Py(V2) Pz(V2) |
| Px(V3) Py(V3) Pz(V3) |

where V is

| V1 |
| V2 |
| V3 |

@GWOMaths @MathsTim See if you can calculate this matrix now.
All the components are functions on (x,y,z),
and the lower left component, Px(V3), is just

Px(V3)
= Px(x^3 + y^3 + z^3)
= Px(x^3) + Px(y^3 + z^3)
= 3.x^2 + 0
= 3.x^2

@GWOMaths @MathsTim When you have evaluated the Jacobean Matrix as above, evaluate it at the point (3,1,2) to obtain a simple 3x3 numeric matrix.

This matrix is the best linear approximation,
at the point (3,1,2)
of our original function.

It is very useful for local calculations, like a town map.

@GWOMaths @MathsTim Only in very unusual circumstances would one worry about the curvature of the Earth when drawing the town map for a community of 20,000 or 30,000 people.

The determinant of a square matrix is a single real number encoding some of the information of the original matrix.

@GWOMaths @MathsTim When the original matrix was the Jacobean Matrix (of partial derivatives), the determinant is termed the Jacobean Determinant. That is the goal of today's problem.

@GWOMaths @MathsTim The formula for evaluating the determinant of a square matrix

| a b c |
| d e f |
| g h i |

is

D = a.(ei - fh) - b.(di - fg) + c.(dh - eg)

Now try it out and see what answer you get.

@GWOMaths @MathsTim @threadreaderapp unroll