• Title: Vector Fields and Potential Functions

  • Series: Multivariable Calculus

  • YouTube-Title: Multivariable Calculus 14 | Vector Fields and Potential Functions

  • Bright video: https://youtu.be/IlkvOqOOgwM

  • Dark video: https://youtu.be/WVEK15dNxi4

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: mc14_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ be the function given by $f(x_1, x_2) = 2 x_1 x_2^7$. What is the gradient of $f$?

    A1: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 14 x_1 x_2^6 \end{pmatrix}$$

    A2: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 1 \ 14 x_1 x_2^6 \end{pmatrix}$$

    A3: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_1^7 \ x_1 x_2^6 \end{pmatrix}$$

    A4: $$ \mathrm{grad}(f)(x_1, x_2) = \begin{pmatrix} 2 x_1 x_2\end{pmatrix}$$

    Q2: Consider the vector function $v: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ given by $$v(x_1, x_2) = \begin{pmatrix} 2 x_2^7 \ 14 x_1 x_2^6 \end{pmatrix}$$ Does $v$ have a potential function?

    A1: Yes, infinitely many ones.

    A2: Yes, but only one.

    A3: No, there is no potential function.

    A4: One needs more information.

    Q3: What is the necessary condition for $v: \mathbb{R}^n \rightarrow \mathbb{R}^n$ having a potential function $f \in C^2(\mathbb{R}^n)$?

    A1: $v$ is continuously differentiable with $$ \frac{\partial v_i}{\partial x_j}(x) = \frac{\partial v_j}{\partial x_i}(x) $$ for all $x, i, j$.

    A2: $v$ is two-times continuously differentiable with $$ \frac{\partial v_i}{\partial x_j}(x) = \frac{\partial v_i}{\partial x_i}(x) $$ for all $x, i, j$.

    A3: $v$ is two-times continuously differentiable with $$ \frac{\partial v_j}{\partial x_j}(x) = \frac{\partial v_i}{\partial x_i}(x) $$ for all $x, i, j$.

    A4: $v$ is continuously differentiable with $$ \frac{\partial v_i}{\partial x_i}(x) = \frac{\partial v_j}{\partial x_j}(x) $$ for all $x, i, j$.

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