# geometric measure theory – Normal and locally normal currents

I am reading the Geometric Measure Theory book by H. Federer and I have some questions about currents:

1. Assuming $$T in mathscr{D}_{m}(U),$$ we call $$T$$ locally normal if and only if $$T$$ is representable by integration and either $$partial T$$ is representable by integration or $$m=0 .$$ Furthermore call $$T$$ normal if and only if $$T$$ is locally normal and spt $$T$$ is compact.

Now, I want to construct a locally normal current that is not normal! Does the following current work:

$$mathbf{E}^{n} llcorner psi$$ corresponding to all weakly differentiable real-valued functions $$psi,$$ with
begin{aligned} left(partialleft(mathbf{E}^{n} llcorner psiright)right)(phi) &=intleftlangle e_{1} wedge cdots wedge e_{n}, psi(x) d phi(x)rightrangle dleft(begin{array}{l} n \ z end{array}right) \ &=(-1)^{n-1} int psi(x) operatorname{div} xi(x) d mathfrak{L}^{n} x end{aligned}

1. $$M subset U$$ oriented submanifold, then there is a corresponding $$n$$-current $$(M)$$ defined by
$$(M)(omega)=int_{M}d H^{n}(x), omega in D^{n}(U)$$

Now, I want to calculate $$partial (M)$$! Is it $$omega$$ on $$M$$??