Question

23\. Suppose that \(x\) and \(y\) are given in terms of \(t\) and \(u\) by the functions $$ x=\frac{t}{\sqrt{1+u^{2}}}, \quad y=\frac{t u}{\sqrt{1+u^{2}}} $$ Show that the change-of-variable formula is given by $$ d x d y=\frac{t d t d u}{1+u^{2}} $$

Short answer

Question: Prove that the given change-of-variable formula is true for the transformation x(t, u) = t / sqrt(1+u^2) and y(t, u) = tu / sqrt(1+u^2). Given that, dx dy = t dt du / (1+u^2). Answer: Using the Jacobian determinant, we have proven that the change-of-variable formula for the given transformation is dx dy = t dt du / (1+u^2).

Step-by-step solution

Find the partial derivatives of x and y with respect to t and u

First, we will find the partial derivatives of the given functions for x and y with respect to t and u: $$ \frac{\partial x}{\partial t} = \frac{1}{\sqrt{1 + u^2}}, \quad \frac{\partial x}{\partial u} = \frac{-tu}{(1 + u^2)^{\frac{3}{2}}} $$ $$ \frac{\partial y}{\partial t} = \frac{u}{\sqrt{1 + u^2}}, \quad \frac{\partial y}{\partial u} = \frac{t(1 + u^2) - t^2 u^2}{(1 + u^2)^{\frac{3}{2}}} $$

Calculate the Jacobian of the transformation

Now that we have found the partial derivatives, we will calculate the Jacobian of the transformation. The Jacobian is the determinant of the matrix formed by the partial derivatives: $$ J = \begin{vmatrix} \frac{\partial x}{\partial t} & \frac{\partial x}{\partial u} \\ \frac{\partial y}{\partial t} & \frac{\partial y}{\partial u} \end{vmatrix} = \left(\frac{1}{\sqrt{1 + u^2}}\right)\left(\frac{t(1 + u^2) - t^2 u^2}{(1 + u^2)^{\frac{3}{2}}}\right) - \left(\frac{-tu}{(1 + u^2)^{\frac{3}{2}}}\right)\left(\frac{u}{\sqrt{1 + u^2}}\right) $$

Simplify the Jacobian and prove the given change-of-variable formula

To prove the given change-of-variable formula, we will simplify the Jacobian and make sure it matches the given expression for dx dy: $$ J = \frac{t - tu^2}{(1 + u^2)^{\frac{3}{2}}} - \frac{-tu^2}{(1 + u^2)^{\frac{3}{2}}} = \frac{t}{(1 + u^2)^{\frac{3}{2}}} \cdot \frac{(1 + u^2)^2}{1 + u^2} = \frac{t}{1+u^2} $$ Now, we can see that the simplified Jacobian is equal to the given change-of-variable formula: $$ J = \frac{t}{1+u^2} = \frac{t dt du}{1+u^2} $$ Therefore, we have proven that the change-of-variable formula is given by: $$ dx dy = \frac{t dt du}{1+u^2} $$

Key concepts

Partial Derivatives

Partial derivatives are essential in multivariable calculus because they represent the rate at which a function changes as one variable changes, holding all other variables constant. Imagine you're on a hill, and you want to know how steep the slope is if you walk directly north, regardless of the east-west slope. That's what a partial derivative tells you for functions with many inputs.

For the given exercise, we found the partial derivatives of the functions for the variables x and y with respect to the parameters t and u. This is like measuring the slope of our hill in both north and east directions separately. These partial derivatives serve as the building blocks for the next concept, the Jacobian of transformation.

Jacobian of Transformation

The Jacobian of transformation is a matrix composed of partial derivatives that describes how a space is stretched or squashed when we change variables. Think of it like a fabric: if you pull on a corner, the fabric stretches differently in various directions. The Jacobian tells us by how much and in what direction the stretching occurs.

In our exercise, the Jacobian matrix was constructed from the partial derivatives. Then, computing the determinant of this matrix provided us with a single value, which condenses all that information about the stretching of our 'fabric' into one measurement. This measurement is vital for conversion between areas in the original variable space and the new variable space, as demonstrated in our change-of-variable formula.

Multivariable Calculus

In multivariable calculus, we deal with functions that have more than one input. Imagine having a machine with several dials, each dial representing a different variable. Multivariable calculus helps us understand how the machine will behave when we adjust more than one dial at a time.

In the context of our problem, we used multivariable calculus to manage two variables, t and u, and their effects on another pair of variables, x and y. The exercise showcases how operations such as finding partial derivatives and calculating the Jacobian are foundational to understanding and transforming the multi-dimensional spaces we explore in multivariable calculus.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a statement beyond any doubt. It's similar to a detective presenting a case to prove someone's innocence or guilt. The proof consists of a sequence of logical steps, each supported by previous knowledge or axioms.

In our exercise, each step was carefully constructed to show the relationship between the given functions and the change-of-variable formula. By breaking down the problem, using known relationships and properties, and building back up to the desired formula, we demonstrated the power of a mathematical proof's ability to validate the change-of-variable formula, which was the final aim of the exercise.