Let’s say some color component, say red, is a linear function of pixel position (x, y) in some region: rᵢ = a₀xᵢ + b₀yᵢ + r₀. This is an arbitrarily oriented linear gradient. Now, if we α-blend this red rᵢ with some other red rⱼ painted over it with some opacity α, we get rₖ = (1 - α)rᵢ + αrⱼ. This gives us a new gradient:
rₖ = (1 - α)(a₀xᵢ + b₀yᵢ + r₀) + α(a₁xᵢ + b₁yᵢ + r₁)
= ((1 - α)a₀ + αa₁)xᵢ + ((1 - α)b₀ + αb₁)yᵢ + (1 - α)r₀ + αr₁.
That is, the three components of the gradient (r, a, b) blend according to the alpha-blending equation just as if they were colors. Except really they are nine components; in some sense a linear gradient is a transformation matrix from (x, y, 1) space into (r, g, b) space. These matrices then react to alpha-blending in the same way that individual color vectors do: (1 - α)M + αN, at least if this α is constant and not a gradient as well.
This means that if we have several layers, each of which is divided into disjoint regions, each painted with a linear gradient with a constant alpha, we can compute the intersection regions of the single-layer regions in different layers, and compute the resulting gradient for each of those intersection regions, then render the resulting gradient.
For things like drop shadows, it would be nice to have gradients of alpha, as well, so that things can be gradually feathered into complete transparency. Linear alpha multiplied by a linearly varying color gives us a quadratic gradient.