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The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in the \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows:
The initial alphabet, $\mathfrak{A} = \{ ``a_1", \ldots, ``a_n" \},$ is extended by a \textit{first order differential alphabet}, $\operatorname{d}\mathfrak{A} = \{ ``\operatorname{d}a_1", \ldots, ``\operatorname{d}a_n" \},$ resulting in a \textit{first order extended alphabet}, $\operatorname{E}\mathfrak{A},$ defined as follows:
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The initial basis, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ is extended by a \textit{first order differential basis}, $\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},$ resulting in the \textit{first order extended basis}, $\operatorname{E}\mathcal{A},$ defined as follows:
The initial basis, $\mathcal{A} = \{ a_1, \ldots, a_n \},$ is extended by a \textit{first order differential basis}, $\operatorname{d}\mathcal{A} = \{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \},$ resulting in a \textit{first order extended basis}, $\operatorname{E}\mathcal{A},$ defined as follows:
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$\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.$
$\operatorname{E}\mathcal{A} = \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} = \{ a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}.$
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The initial space, $A = \langle a_1, \ldots, a_n \rangle,$ is extended by a \textit{first order differential space} or \textit{tangent space}, $\operatorname{d}A = \langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle,$ at each point of $A$, resulting in a \textit{first order extended space} or \textit{tangent bundle}, $\operatorname{E}A,$ defined as follows:
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$\operatorname{E}A = A\ \times\ \operatorname{d}A = \langle \operatorname{E}\mathcal{A} \rangle = \langle \mathcal{A}\ \cup\ \operatorname{d}\mathcal{A} \rangle = \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.$
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