Here you find exercises and solutions for multivariable calculus.
Part 1 - Gradient
Let’s calculate the gradient of a function defined on $\mathbb{R}^3$. A typical application of that is found in physics, more precisely in electrodynamics. The electrostatic potential of a point charge can be modelled by such a function and the gradient represents the electric field.
Part 2 - Maxima and Minima
Let’s got to a two-dimensional example where we can calculate the gradient and the Hessian. Afterwards, we want to find all maxima and minima by looking at the critical points and the eigenvalues of the Hessian.
Part 3 - Jacobian Matrix
The Jacobian matrix represents the derivative of a map $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$. Let’s look at $3\times 3$-example and let’s also calculate the determinant in this case.
Part 4 - Taylor Polynomial in One Dimension
In order to understand the Taylor polynomial in higher dimensions, we have to consider a one-dimensional example first.
Part 5 - First Order Taylor Polynomial in Two Dimensions
The Taylor polynomial in higher dimensions is not really more complicated; one just have to calculate more partial derivatives. Let’s demonstrate this for a function of two variables $x$ and $y$, which we call sombrero function because of the looks of the graph.
Part 6 - Second Order Taylor Polynomial in Two Dimensions
Summary of the course Exercises - Multivariable Calculus
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