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Your u variable is a symbolic expression, and the function call syntax with positional/unnamed arguments -- that you use when you write u(x,y,z,t) -- is deprecated, exactly because it is confusing.

You should write instead

d1 = diff(u,x,t,t,z)
d2 = diff(v,z,z,t,t)

Also, you defined x,y,z,t as generators of a polynomial ring, but when you substitute them into a symbolic expression (like your symbolic function) they are coerced to symbolic variables of the same name, so you might as well have defined var('x,y,z,t').