Question

Calculate the Laplacian ${\mathrm{\nabla }}^2$ of each of the following scalar fields. $$\sqrt{x^2-y^2}$$

Short answer

The Laplacian of $\sqrt{x^2-y^2}$ is $\frac{1}{\sqrt{x^2-y^2}}$.

Step-by-step solution

Write down the Laplacian operator in Cartesian coordinates

The Laplacian in Cartesian coordinates is given by $$abl{a}^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}$$ We will only consider the variables present in the scalar field.

Calculate the first partial derivatives

Compute the first partial derivatives of $f=\sqrt{x^2-y^2}$ with respect to $x$ and $y$. $$\frac{\partial f}{\partial x}=\frac{x}{\sqrt{x^2-y^2}}$$$$\frac{\partial f}{\partial y}=\frac{-y}{\sqrt{x^2-y^2}}$$

Compute the second partial derivatives

Calculate the second partial derivatives of $f$ with respect to $x$ and $y$. For $x$: $$\frac{{\partial }^{2}f}{\partial x^2}=\frac{(x^2-y^2)-x^2}{(x^2-y^2{)}^{3/2}}=\frac{-y^2}{(x^2-y^2{)}^{3/2}}$$For $y$:$$\frac{{\partial }^{2}f}{\partial y^2}=\frac{(x^2-y^2)-(-y^2)}{(x^2-y^2{)}^{3/2}}=\frac{x^2}{(x^2-y^2{)}^{3/2}}$$

Sum the second partial derivatives

Add the second partial derivatives to find the Laplacian of the scalar field. $$abl{a}^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}=\frac{-y^2}{(x^2-y^2{)}^{3/2}}+\frac{x^2}{(x^2-y^2{)}^{3/2}}=\frac{x^2-y^2}{(x^2-y^2{)}^{3/2}}=\frac{1}{\sqrt{x^2-y^2}}$$

Key concepts

Partial Derivatives

Partial derivatives are like normal derivatives, but they focus on one variable while keeping others constant. This is essential when working with functions of multiple variables.
For example, if you have a function $f(x,y)$, taking the partial derivative with respect to $x$ means you treat $y$ as a constant.

• The notation for the partial derivative of $f$ with respect to $x$ is $\frac{\text{textpartdiff}f}{\text{textpartdiff}x}$.
• Similarly, the partial derivative with respect to $y$ is $\frac{\text{textpartdiff}f}{\text{textpartdiff}y}$.

Using partial derivatives, you can determine how a function changes as each variable changes.
For example, for the function $f(x,y)=\text{textsqrt}{x}^2-y^2$, the partial derivatives are:
\frac{\text{\textpartdiff} f}{\text{\textpartdiff} x} = \frac{x}{\textsqrt{x^2 - y^2}} and \frac{\text{\textpartdiff} f}{\text{\textpartdiff} y} = \frac{-y}{\textsqrt{x^2 - y^2}}. These results help us understand how $f$ behaves with different variable changes.

Scalar Fields

A scalar field assigns a single value (a scalar) to every point in a space. In simpler terms, it's a mathematical function where you only get one output value no matter how many input variables you provide.
Common examples include temperature distribution in a room or elevation on a map.
In our exercise, we are dealing with the scalar field $f(x,y)=\text{textsqrt}{x}^2-y^2$.
Each point $(x,y)$ in this field gets a scalar value given by the function.
Understanding scalar fields is vital because they are used extensively in physics, engineering, and other sciences to describe various phenomena. They make it easier to represent quantities that vary over space, like gravitational or electric fields.

Cartesian Coordinates

Cartesian coordinates are used to specify any point in a 2D plane or 3D space using numerical values. These points are defined by ordered pairs (x, y) in 2D or triplets (x, y, z) in 3D.

• The $x$-axis is typically the horizontal axis.
• The $y$-axis is the vertical axis in 2D, and in 3D, it's the depth.
• The $z$-axis represents height in 3D space.

Cartesian coordinates are convenient for solving mathematical problems because they form a grid.
In our exercise, we are working with a 2D Cartesian coordinate system, specifically on the function $f(x,y)$. Understanding this system makes it easier to compute partial derivatives and ultimately find the Laplacian.

Laplacian Operator

The Laplacian operator, denoted as ${\text{abla}}^2$, is essential in mathematics and physics for finding how a scalar field changes in all spatial directions.
It’s especially useful in solving equations involving physical phenomena like heat flow, wave propagation, or electrostatics.
In Cartesian coordinates for a function $f(x,y,z)$, the Laplacian operator is:
\text{abla}^2 = \frac{\text{\textpartdiff}^2}{\text{\textpartdiff} x^2} + \frac{\text{\textpartdiff}^2}{\text{\textpartdiff} y^2} + \frac{\text{\textpartdiff}^2}{\text{\textpartdiff} z^2} Our exercise uses a 2D function, so we adapt the formula:
\text{abla}^2 f = \frac{\text{\textpartdiff}^2 f}{\text{\textpartdiff} x^2} + \frac{\text{\textpartdiff}^2 f}{\text{\textpartdiff} y^2} By computing second partial derivatives and summing them, we obtain the Laplacian. This helps us understand how the field behaves spatially. For $f(x,y)=\text{textsqrt}{x}^2-y^2$, the Laplacian simplifies to $\frac{1}{\text{textsqrt}{x}^2-y^2}$, showing us the rate of change in the field.