position vector

In the space $ \mathbb{R}^3$ , the vector

$\displaystyle \vec{r} \;:=\; (x,\,y,\,z) \;=\; x\vec{i}+y\vec{j}+z\vec{k}$

directed from the origin to a point $ (x,\,y,\,z)$ is the position vector of this point. When the point is variable, $ \vec{r}$ represents a vector field and its length

$\displaystyle r \;:=\; \sqrt{x^2+y^2+z^2}$

a scalar field.

The simple formulae

are valid, where $ \vec{r}^0$ is the unit vector having the direction of $ \vec{r}$ .

If $ \vec{c}$ is a constant vector, $ \vec{U}\!\!:\mathbb{R}^3\to\mathbb{R}^3$ a vector function and $ f\!\!:\mathbb{R}\to\mathbb{R}$ is a twice differentiable function, then the formulae

hold.

Bibliography

1
K. V¨AISÄLÄ: Vektorianalyysi. Werner Söderström Osakeyhtiö, Helsinki (1961).



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