If you have one function that takes a single number to a high dimensional space, and another which maps that high dimensional space back down to the number line, applying one after the other gives a regular old single-variable function. But how do you find the derivative of this new function?
Test your knowledge of the various ways to extend the idea of a derivative to multivariable function.
The Laplacian, also known as the "Laplace operator", is a way to extend the second derivative to multivariable functions.
In the fluid flow interpretation of vector fields, divergence is a measure of how much fluid tends to flow away from each point. However, his turns out to have far-reaching consequences beyond the specific case of fluid flow.
How do you measure how much a curve actually curves? There are many ways to view the answer to this question, but all land you on the same mathematical quantity, known simply as "curvature".
Divergence and curl are two ways of extending the idea of a derivative to vector fields. If you interpret a vector field as representing a fluid flow, the divergence tells you if the fluid tends to converge near a given point, or if it diverges away from it. The curl, on the other hand, measures rotation in the fluid.
Differentiating_vector_valued_functions_articles_
What it means to take the derivative (or partial derivative) of a vector-valued function
Curl is a special version of the derivative for vector fields. It can be nicely interpreted in terms of rotation in fluid flow.
The meaning of the derivative is not so straightforward when the input space has multiple dimensions. Partial derivatives allow us to handle this by restricting movement in the input space.
Gradient_and_directional_derivatives
The gradient is like the king of all partial derivatives. Or perhaps it's more like the country they all live it. It stores all partial derivative information in a single vector-valued function, and as such it is a central tool for analyzing rates of change in multivariable functions.
Differentiating_parametric_curves
In this tutorial, we will explore position vector functions and think about what it means to take a derivative of one.
Partial_derivative_and_gradient_articles_
There are several different analogs of the derivative for scalar-valued multivariable functions: Partial derivatives, the gradient, and the directional derivative.
Partial_derivatives_of_vector_valued_functions
Learn how to compute and interpret the partial derivative of a multivariable function with a vector-valued output.
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