Calculus Flashcards
What is the difference between
Δy, 𝑑y, ∂y
All three are ways of conceptualizing the idea of a difference of values.
All of them assume that y is, at least implicitly, a function of one or more variables, so it is possible to talk about y at “two different places” to find the difference between them
Δy represents the difference in y when evaluated at two different points. For instance, to calculate the slope of a line, you have y(p), where p is a point and y is (conventionally) the y-coordinate of that point in some Cartesian coordinate system. In that situation, Δy would refer to y(p2)−y(p1), for some given pair of points.
Δy is also used when y=(y1,y2,y3,…) is a sequence of values, and Δy would then refer to the “sequence of differences” Δy=(y2−y1,y3−y2,…,yn+1−yn,…). This is used in the “calculus of finite differences”, which is a discrete analog of the normal calculus. In this case, Δy acts as a function (from integers) in much the same way that y does.
dy represents the difference in 𝑦y when evaluated at two points which are infinitesimally close — or, if you want to get technical, the limit of Δy when the difference between the points goes to zero. Usually, this differential is used in a ratio with a different differential, to get a derivative like dy/dx = lim Δx→0 for Δy/Δx. dy is commonly used in single-variable calculus, but it has other uses (which I’ll mention later on). The derivative of a function is another function, so like in the finite differences case, differentiation takes functions to functions.
∂y is used when 𝑦y depends on multiple variables, and it is always used in reference to an explicit variable, as in ∂y/∂x. This usage implies that y(x,z,…,w) is a function of multiple variables, not just x. It refers to the partial differential, in much the same way as dy refers to the (total) differential. It is defined basically as “take the derivative dy/dx, assuming z,…,w are all held constant”.
Partial derivatives imply a “total” derivative, which is a combination of all the partial derivatives in a sensible way. The total derivative is represented by dy, bringing us back to the previous notation. It’s worth mentioning that a partial derivative of a single-variate function is the same as the total derivative, so the use of ∂y only occurs when doing multivariate calculus.
What is the difference between
∇y , ∇⋅y , ∇×y
You also have ∇, which is a multivariable differential operator representing the “vector” ∇y = ∂y/∂x, ∂y/∂z, ∂y/∂w), and is used in various ways on multivariate functions.
1) If y returns a real number, then ∇y is the gradient of y, a vector-valued multivariate function that points in the direction of greatest change
2) If y is vector-valued, then ∇⋅y is the divergence of y, a single value multivariate function that relates to how much is “flowing out” of a given point.
3) There’s also ∇×y, which is the curl of y, and is a vector-valued multivariate function that shows how much the function y is “flowing around” a given point.
