Before the vector calculus was invented mathematicians used to do vector calculus like operations using quaternions. Quaternions are four dimensional extension of complex numbers. They have two additional imaginary units $j$ and $k$. Quaternion multiplication is non-commutative and is defined by the formula:

$$i^2=j^2=k^2=ijk ...

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I was never good with remembering if derivate of $\sin x$ was plus or minus $\cos x$. Or if the derivative of $\cos x$ was plus or minus $\sin x$. I could just remember that one derivative had minus and other didn't. But I don't have that problem ...

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