Where can I find a proof that the definition of total variation for functions of several variables is consistent with the definition of total variation for functions of one variable?
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1$\begingroup$ This is addressed e.g. in the book Functions of Bounded Variation by L. Ambrosio, N. Fusco and D. Pallara (2000). See e.g. section 3.2 therein. $\endgroup$– SkeeveCommented Apr 13, 2019 at 21:04
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$\begingroup$ @Skeeve I couldn't find it. What proposition is it? $\endgroup$– RikuCommented Apr 13, 2019 at 21:58
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2$\begingroup$ you are right, in that book it is stated just one way in Theorem 3.28 (see the discussion of good representatives). But if you want an explicit statement about equivalence of the two definitions then have a look at Theorem 1 in section 5.10.1 of Measure theory and fine properties of functions by L.C. Evans and R.F. Gariepy (1992). $\endgroup$– SkeeveCommented Apr 14, 2019 at 6:36
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$\begingroup$ @Skeeve Found it. That's great. Thank you. $\endgroup$– RikuCommented Apr 20, 2019 at 14:53
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