I am trying to complete a problem where I have to show that the given operator is not linear: $L[f] = \partial_t f + f\partial_x f$ I currently am using the superposition principle to formally show that it is in fact non linear, this is what I have currently $L[f] = \partial_t f + f\partial_x f\\ L[\alpha u + \beta v] = [\alpha u + \beta v]\partial_t + [\alpha u + \beta v]\partial_x[\alpha u + \beta v]\\ =\partial_t\alpha u + \partial_t\beta v + [\alpha u + \beta v][\partial_x \alpha u + \partial_x \beta v]\\ = \partial_t\alpha u + \partial_t\beta v + \alpha u\partial_x \alpha u + \alpha u \partial_x \beta v + \beta v \partial_x \alpha u + \beta v \partial_x \beta v\\ = \alpha[\partial_t u + u\partial_x u] + \beta[\partial_t v + v\partial_x v] + \alpha u \partial_x \beta v + \beta v \partial_x \alpha u$
So this is non linear as we have the additional terms on the end. Is this correct?