Vector calculus and partial derivatives from "summary" of Differential Calculus by S Balachandra Rao
Vector calculus is a branch of mathematics that deals with vectors, which are quantities that have both magnitude and direction. When applied to calculus, it involves the study of functions of several variables, where each variable represents a different direction in space. Partial derivatives play a crucial role in vector calculus, as they allow us to understand how a function changes with respect to each of its variables, while holding the other variables constant. By taking partial derivatives, we can calculate the rate of change of a function in a specific direction, which is essential for solving problems in physics, engineering, and other fields. In vector calculus, we often work with vector-valued functions, which are functions that output vectors instead of scalars. These functions can be differentiated component-wise, giving us a way to understand how each component of the function changes independently. This approach is particularly useful when dealing with vector fields, where a vector is assigned to each ...
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