1. vector field

Let$\displaystyle f:R^3\to R$ be a scalar field, $\displaystyle V:R^3 \to R^3$ be a vector field and $\displaystyle a\in R^3$ be a constant vector. If $\displaystyle r$ represents the position vector $\displaystyle xi+yj+zk$, then which of the following in not true? (Justify)
(1) $\displaystyle Curl(fv)=grad(f) \times v+fCurl(v)$
(2)$\displaystyle div(grad(f))=\big(\frac{\delta^2}{\delta x^2}+\frac{\delta^2}{\delta y^2}+\frac{\delta^2}{ \delta z^2} \big)f$
(3)$\displaystyle Curl(a \times r)=2|a|r$
(4) $\displaystyle div(\frac{r}{|r|^3})=0; for \ r \ne 0$

3. Re: vector field

Originally Posted by kjchauhan

Let$\displaystyle f:R^3\to R$ be a scalar field, $\displaystyle V:R^3 \to R^3$ be a vector field and $\displaystyle a\in R^3$ be a constant vector. If $\displaystyle r$ represents the position vector $\displaystyle xi+yj+zk$, then which of the following in not true? (Justify)
(1) $\displaystyle Curl(fv)=grad(f) \times v+fCurl(v)$
(2)$\displaystyle div(grad(f))=\big(\frac{\delta^2}{\delta x^2}+\frac{\delta^2}{\delta y^2}+\frac{\delta^2}{ \delta z^2} \big)f$
(3)$\displaystyle Curl(a \times r)=2|a|r$
(4) $\displaystyle div(\frac{r}{|r|^3})=0; for \ r \ne 0$
I find this notation confusing. Why are you using both $V~\&~v$? Are they meant to be the same vector field.
In #2, Does $\big(\frac{\delta^2}{\delta x^2}+\frac{\delta^2}{\delta y^2}+\frac{\delta^2}{ \delta z^2} \big)f$ taking second derivatives of $f~?$