partial-derivative notation

Before you can read a PDE you have to read its derivatives, and the subject uses a compact shorthand that takes ten minutes to learn and saves you forever. A partial derivative is the slope of u in one chosen direction with the other variables frozen. The fully spelled-out symbol is the curly-d notation, partial u over partial x, but writing that out for every term is exhausting, so almost everyone uses subscripts: u_x means the partial derivative of u with respect to x, u_t with respect to t, and so on.

Higher and mixed derivatives stack the subscripts. u_xx is the partial derivative of u_x again with respect to x (a second derivative, measuring curvature in the x-direction); u_tt is the second time derivative; u_xy means differentiate first by x then by y. For smooth functions the order of mixed differentiation does not matter, so u_xy and u_yx are the same — a small fact (Clairaut's theorem) that you lean on constantly. Three more bundles appear so often they get their own names: the gradient grad u (written nabla u) collects all the first space derivatives into a vector (u_x, u_y, u_z) pointing in the direction u increases fastest; the divergence div of a vector field F = (P, Q, R) is the scalar P_x + Q_y + R_z, measuring net outflow; and the Laplacian, Laplacian u (also written nabla^2 u or Delta u), is div grad u = u_xx + u_yy + u_zz.

These symbols are not decoration — they encode the geometry. A first derivative is a rate or a slope; a second derivative like u_xx is a curvature; the gradient is a direction of steepest ascent; the divergence is 'how much is leaving this point'; the Laplacian compares u at a point to its average nearby. Once you can glance at u_t = k u_xx and hear 'rate of change in time equals diffusivity times curvature in space', the equations start to mean something rather than just sit on the page.