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Question about proof in a paper using fundamental theorem of calculus
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I'm reading a paper "Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation" https://arxiv.org/abs/1705.08475
There, I try to understand proof for Theorem 2.1. It starts with saying "By the main theorem of calculus, it holds that:"
$$
f_j(x+delta) = f_j(x) + int_0^1left<nabla f_j(x+tdelta),deltaright> d t, text{ for } j = 1,dots,K.
$$
Where, $f_j(x): mathbb{R}^d rightarrow mathbb{R}$, $delta in mathbb{R}^d$
A simple way to formulate the fundamental theorem of calculus would be:
$$
f_j(x+delta) = f_j(x) + left<nabla f_j(x+delta),deltaright> , text{ for } j = 1,dots,K.
$$
But I do not understand where the integral comes from, and why is it correct?
calculus proof-explanation
asked Jan 13 at 13:13
mcsimmcsim
1578
$endgroup$
add a comment |
$begingroup$
I'm reading a paper "Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation" https://arxiv.org/abs/1705.08475
There, I try to understand proof for Theorem 2.1. It starts with saying "By the main theorem of calculus, it holds that:"
$$
f_j(x+delta) = f_j(x) + int_0^1left<nabla f_j(x+tdelta),deltaright> d t, text{ for } j = 1,dots,K.
$$
Where, $f_j(x): mathbb{R}^d rightarrow mathbb{R}$, $delta in mathbb{R}^d$
A simple way to formulate the fundamental theorem of calculus would be:
$$
f_j(x+delta) = f_j(x) + left<nabla f_j(x+delta),deltaright> , text{ for } j = 1,dots,K.
$$
But I do not understand where the integral comes from, and why is it correct?
calculus proof-explanation
asked Jan 13 at 13:13
mcsimmcsim
1578
$endgroup$
add a comment |
$begingroup$
I'm reading a paper "Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation" https://arxiv.org/abs/1705.08475
There, I try to understand proof for Theorem 2.1. It starts with saying "By the main theorem of calculus, it holds that:"
$$
f_j(x+delta) = f_j(x) + int_0^1left<nabla f_j(x+tdelta),deltaright> d t, text{ for } j = 1,dots,K.
$$
Where, $f_j(x): mathbb{R}^d rightarrow mathbb{R}$, $delta in mathbb{R}^d$
A simple way to formulate the fundamental theorem of calculus would be:
$$
f_j(x+delta) = f_j(x) + left<nabla f_j(x+delta),deltaright> , text{ for } j = 1,dots,K.
$$
But I do not understand where the integral comes from, and why is it correct?
calculus proof-explanation
asked Jan 13 at 13:13
mcsimmcsim
1578
$endgroup$
I'm reading a paper "Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation" https://arxiv.org/abs/1705.08475
There, I try to understand proof for Theorem 2.1. It starts with saying "By the main theorem of calculus, it holds that:"
$$
f_j(x+delta) = f_j(x) + int_0^1left<nabla f_j(x+tdelta),deltaright> d t, text{ for } j = 1,dots,K.
$$
Where, $f_j(x): mathbb{R}^d rightarrow mathbb{R}$, $delta in mathbb{R}^d$
A simple way to formulate the fundamental theorem of calculus would be:
$$
f_j(x+delta) = f_j(x) + left<nabla f_j(x+delta),deltaright> , text{ for } j = 1,dots,K.
$$
But I do not understand where the integral comes from, and why is it correct?
calculus proof-explanation
calculus proof-explanation
asked Jan 13 at 13:13
mcsimmcsim
1578
asked Jan 13 at 13:13
mcsimmcsim
1578
asked Jan 13 at 13:13
mcsimmcsim
1578
asked Jan 13 at 13:13
mcsimmcsim
1578
asked Jan 13 at 13:13
mcsimmcsim
1578
1578
add a comment |
add a comment |
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