MS-A0211 - Differential and Integral Calculus 2, Lecture, 10.1.2023-20.2.2023
This course space end date is set to 20.02.2023 Search Courses: MS-A0211
Lecture 5 - 24.1.2023
Completion requirements
In this lecture we studied:
- Linear approximations and differentiability. We reviewed linear approximation and derivatives in one variable. Analogously, we defined the linear approximation for a function of 2 variables (i.e., the tangent plane), and we introduced the notion of differentiability for a function of 2 variables, by assuming that the partial derivatives exist. Pointed out that the condition of differentiability says that the linear approximation (tangent plane) "works" in the sense that the relative error goes to zero as the initial point is approached. We stated and proved the fact that differentiability implies continuity. For that, we used Squeeze theorem. We also stated the fact that continuous partial derivatives implies differentiability.
- Gradient vector. Directional derivatives. We introduced the gradient vector at a point for a function f of 2 variables. Then, we recalled the notion of directional derivative D_v f of f with respect to a vector v, whether v is a unit vector or not. If v is a unit vector, then D_v f is the slope of the surface in the given direction v. We showed how D_v f can be computed by using the gradient vector of f, by assuming that the partial derivatives exist. We noticed the relationship between the directional derivative of a vector and the directional derivative of the corresponding normalized vector. The speed is the conversion factor between these measurements. We used directional derivatives to study the rate of change of a function in any given direction. In particular, we studied the direction of maximum increase of a function and the direction of maximum decrease.
- Gradient vector and level curves. We proved that the rate of change is zero in the directions tangent to the level curve of the function at the corresponding point. We also showed that the gradient vector is orthogonal to the level curve.
The notions studied in this lecture can be found in sections 12.6 and 12.7 of Adam and Essex’s Calculus book (seventh edition).
Maple codes of the function gradplot (two dimensional vector-field with the gradient vectors)