Dot product

Theory and Concepts
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1

PKB please Explain how to use the dot product to extract one component of a vector equation such as the Navier-Stokes equations or the integral conservation of momentum equation

2

This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
Below is a concise explanation.

  1. Begin with a vector equation such as the integral conservation of momentum:
\frac{\partial}{\partial t}\int_{\mathcal{V}} \rho\,\vec{u}\,dV \;+\; \int_{\mathcal{S}} \left(\rho\,\vec{u}\right) \left(\hat{n}\cdot\vec{u}\right)dA \;=\; \int_{\mathcal{S}} -p\,\hat{n}\,dA \;+\; \int_{\mathcal{V}} -\rho\,g\,\hat{z}\,dV\,.
  1. To extract, for example, the x-component of this equation, take the dot product of every term with the unit vector \hat{x}. That is, apply
\big[\cdots\big] \cdot \hat{x}\,.
  1. Use the linearity of the dot product. For instance, note that time differentiation and integration commute with the dot product. Thus, the equation becomes
\frac{\partial}{\partial t}\int_{\mathcal{V}} \rho\,\vec{u}\cdot\hat{x}\,dV \;+\; \int_{\mathcal{S}} \left(\rho\,\vec{u}\, (\hat{n}\cdot\vec{u})\right)\cdot \hat{x}\,dA \;=\; \int_{\mathcal{S}} \left(-p\,\hat{n}\right)\cdot \hat{x}\,dA \;+\; \int_{\mathcal{V}} \left(-\rho\,g\,\hat{z}\right)\cdot \hat{x}\,dV\,.

  1. Evaluate each dot product term:

    • For the first term, \vec{u}\cdot\hat{x} is simply the x-component of the velocity, denoted by u.

    • In the second term, (\rho\,\vec{u}) projects to \rho u when dotted with \hat{x}, while the factor (\hat{n}\cdot\vec{u}) remains a scalar.

    • In the pressure term, (-p\,\hat{n})\cdot \hat{x} = -p\,(n_x), where n_x is the x-component of the boundary normal vector.

    • The gravitational term vanishes because \hat{z}\cdot\hat{x} = 0.

  2. The simplified scalar equation for x momentum is then:

\frac{\partial}{\partial t}\int_{\mathcal{V}} \rho\,u\,dV \;+\; \int_{\mathcal{S}} \rho\,u\,(\hat{n}\cdot\vec{u})\,dA \;=\; \int_{\mathcal{S}} -p\,n_x\,dA\,.
  1. In summary, the dot product with \hat{x} extracts the x-component from each vector term. In the original equation, the terms were vectors or dyadic products; after taking the dot product with \hat{x}, they collapse into scalar quantities describing only the x momentum.

The process is general: select a unit vector in the desired direction, apply the dot product term by term, and use the fact that differentiation and integration commute with the dot product to arrive at a scalar conservation law for that specific component.

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked their significance below. If they are marked as relevant, I encourage you to explore them:

Textbook sections:

  1. Appendix F.3: Volumetric Viscous Force Part 1 (RELEVANT)
  2. Chapter 3.5: Hydrostatic Equation: Vector Differential Operator Form (RELEVANT)
  3. Chapter 7.3: Integral Conservation of Momentum Part 1 (RELEVANT)
  4. Appendix B.4: Vector Calculus Part 3 (RELEVANT)
  5. Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (RELEVANT)

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