Unit vector notation

Textbook
1

my math textbooks use \hat{i} \hat{j} \hat{k}, why does your textbook not do that

2

PK Bot’s answer:
In fluid mechanics, the notation used can vary depending on the context and the preferences of the author. In Professor Kirby’s course material, the notation for unit vectors is \boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}, and \boldsymbol{\hat{z}} for Cartesian coordinates. This is a stylistic choice and is consistent with the conventions used in fluid mechanics, where vectors are often denoted with boldface and unit vectors with a circumflex.

The use of \hat{i}, \hat{j}, and \hat{k} is common in many mathematics and physics textbooks, particularly in the context of vector calculus and introductory physics. These symbols represent the unit vectors in the Cartesian coordinate system, similar to \boldsymbol{\hat{x}}, \boldsymbol{\hat{y}}, and \boldsymbol{\hat{z}}.

The choice of notation does not affect the underlying mathematics or physics; it is simply a matter of convention. In fluid mechanics, the notation \boldsymbol{\vec{u}} is often used for velocity, and the unit vectors are denoted with a circumflex to distinguish them from other vector quantities. This helps maintain consistency and clarity, especially when dealing with complex vector and tensor operations.

To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (HIGHLY RELEVANT)
  2. Appendix B.1: Vector Algebra Part 1 (HIGHLY RELEVANT)
  3. Chapter 3.5: Hydrostatic Equation: Vector Differential Operator Form (HIGHLY RELEVANT)
  4. Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
  5. Appendix B.4: Vector Calculus Part 3 (HIGHLY RELEVANT)

Disclaimer:

See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.

3

I don’t always use Cartesian coordinates; i, j, and k are fine I suppose if you always use Cartesian coordinates. If you use other coordinates you find that you want to use symbols that denote the coordinate, for example \hat{r} and \hat{\theta} for cylindrical. So if r and \theta are denoted by \hat{r} and \hat{\theta}, then why not use \hat{x}, \hat{y}, \hat{z}. Some people use e_x, e_y, e_z; I think that notation is useful if you are generalizing especially in higher-dimensional spaces, but I don’t do that in my fluids class