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of r copies of W, i.e., ''W'' ? ''W'' ? ··· ? ''W''. However, elements of such a tensor product do not pull back naturally: instead there is a pushforward operation from ''V'' ? ''V'' ? ··· ? ''V'' to ''W'' ? ''W'' ? ··· ? ''W'' given by

Nevertheless, it follows from this that if ? is invertible, pullback can be defined using pushforward by the inverse function ?^-1. Combining these two constructions yields a pushforward operation, along an invertible linear map, for tensors of any rank (''r'', ''s'').

Pullback of cotangent vectors and 1-forms

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