• Title: Determinant is a Volume Measure

• Series: Linear Algebra

• Chapter: Determinants

• YouTube-Title: Linear Algebra 45 | Determinant is a Volume Measure

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• Quiz Content

  Q1: Consider a $3$-dimensional volume function $\mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w})$. Which operation is correct?

  A1: $$\mathrm{vol}_3(\mathbf{u}, 2\mathbf{v}, \mathbf{w}) = 2 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A2: $$\mathrm{vol}_3(\mathbf{u}, \mathbf{v} + \mathbf{u}, \mathbf{w}) = 2 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A3: $$\mathrm{vol}_3(2 \mathbf{u}, 2 \mathbf{v}, 2 \mathbf{w}) = 2 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A4: $$\mathrm{vol}_3(2 \mathbf{u}, 2 \mathbf{v}, 2 \mathbf{w}) = 4 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  Q2: Consider a $3$-dimensional volume function $\mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w})$. Which operation is correct?

  A1: $$\mathrm{vol}_3(\mathbf{u}, \mathbf{v} + \mathbf{u} +\mathbf{w}, \mathbf{w}) = \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A2: $$\mathrm{vol}_3(\mathbf{u}, \mathbf{v} + \mathbf{u}, \mathbf{w}) = 2 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A3: $$\mathrm{vol}_3(2 \mathbf{u}, \mathbf{v}, 2 \mathbf{w}) = \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  A4: $$\mathrm{vol}_3(2 \mathbf{u}, 2 \mathbf{v}, 2 \mathbf{w}) = 4 \cdot \mathrm{vol}_3(\mathbf{u}, \mathbf{v}, \mathbf{w}) $$

  Q3: Consider the two-dimensional volume form $\mathrm{vol}_2(\mathbf{u}, \mathbf{v})$. What is the correct formula to calculate it?

  A1: $\mathrm{vol}_2(\mathbf{u}, \mathbf{v}) = u_1 v_2 - v_1 u_2$

  A2: $\mathrm{vol}_2(\mathbf{u}, \mathbf{v}) = u_2 v_2 - v_1 u_1$

  A3: $\mathrm{vol}_2(\mathbf{u}, \mathbf{v}) = u_1 v_1 - v_2 u_2$

  A4: $\mathrm{vol}_2(\mathbf{u}, \mathbf{v}) = u_2 v_1 - v_2 u_1$

• Last update: 2024-10

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