# Physics equations/00-Mathematics for this course

This resources uses, openstax Physics, an open source textbook available for free at http://cnx.org/content/col11406/1.7) Most sections of the Physics equations have a link to the appropriate chapter.

This section is an exception, since openstax Physics has no such review.

### Equations found on Physeq templates

RiemannSum
• ${\displaystyle \int _{a}^{b}f(x)\mathrm {d} x\approx \sum f(x_{j})\Delta x_{j}}$ is the Riemann sum representation of the integral of f(x) from x=a to x=b. It is the area under the curve, with contributions from f(x)<0 being negative (if a>b). The sum equals the integral in the limit that the widths of all the intervals vanish (Δxj→0).
UnitVectors
• A unit vector is any vector with unit magnitude equal to one. For any nonzero vector, ${\displaystyle {\hat {V}}={\vec {V}}/V}$ is a unit vector. An important set of unit vectors is the orthonormal basis associated with Cartesian coordinates:
• ${\displaystyle \mathbf {\hat {i}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {j}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {k}} =1}$
• ${\displaystyle \mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =\mathbf {\hat {k}} \cdot \mathbf {\hat {i}} =\mathbf {\hat {j}} \cdot \mathbf {\hat {k}} =0}$
• The basis vectors ${\displaystyle (\mathbf {\hat {i}} ,\mathbf {\hat {j}} ,\mathbf {\hat {k}} )}$ are also written as ${\displaystyle ({\hat {x}},{\hat {y}},{\hat {z}})}$, so that any vector may be written ${\displaystyle {\vec {A}}=A_{x}{\hat {x}}+A_{y}{\hat {y}}+A_{z}{\hat {z}}}$. Even more elegance is achieved by labeling the directions with integers: ${\displaystyle {\vec {A}}=A_{1}{\hat {e_{1}}}+A_{2}{\hat {e_{2}}}+A_{3}{\hat {e_{3}}}}$ ${\displaystyle =\Sigma A_{j}{\hat {e_{j}}}\,.}$
DotProduct
• ${\displaystyle {\vec {A}}\cdot {\vec {B}}=AB\cos \theta =A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}\,}$ is the dot product between two vectors separated in angle by θ.
CrossProductVisual
• ${\displaystyle {\vec {A}}\times {\vec {B}}={\vec {C}}}$ is the cross product of ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$. The cross product, ${\displaystyle {\vec {C}}}$ is directed perpendicular to ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ by the right hand rule.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C=|AB\sin \theta |}$ wehre ${\displaystyle \theta }$ is the angle between vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle |{\vec {A}}\times {\vec {B}}|=C}$ is also the magnitude of the of the parallelogram defined by the vectors ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$.
• ${\displaystyle {\vec {A}}\times {\vec {B}}=0}$ if ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ are either parallel or antiparallel.
• The unit vectors obey ${\displaystyle {\hat {x}}\times {\hat {y}}={\hat {z}}}$, ${\displaystyle {\hat {y}}\times {\hat {z}}={\hat {x}}}$, and ${\displaystyle {\hat {z}}\times {\hat {x}}={\hat {y}}}$.

## Wikiversity resources

### CALCULUS-based Wikiversity resources

• Vector calculus Vector derivatives. Gradient, div, and stokes theorems.
• Coordinate systems Designed to facilitate a simple understanding of line, surface and volume integrals.
• Coulomb's Law Introduces the Coulomb integral using a line charge. It then extends this result to a plane charge using a double integral.