Multivariable Calculus explores functions of multiple variables, including partial derivatives, multiple integrals, and vector calculus.
In multivariable calculus, we study how a function changes as just one variable varies and the others stay fixed. This leads to partial derivatives.
The partial derivative of \(f(x, y)\) with respect to \(x\) is written as \(\frac{\partial f}{\partial x}\). It measures how fast \(f\) changes as \(x\) changes, holding \(y\) constant.
To compute \(\frac{\partial f}{\partial x}\), treat \(y\) as a constant and differentiate as usual with respect to \(x\).
They help us understand the sensitivity of systems to different variables. They're essential in fields like physics, engineering, and economics.
Just like with single-variable functions, you can take derivatives of derivatives, leading to concepts like the Hessian matrix.
The rate at which the temperature changes as you move east, keeping your north-south position fixed, is a partial derivative.
The speed of a chemical reaction with respect to temperature, holding concentration constant, is found using partial derivatives.
Partial derivatives show how a function changes as each variable changes on its own.