Question

Let $F: mathbb{R}^3 ightarrow mathbb{R}^2$ be a differentiable function. Let $p in mathbb{R}^3$. Use the definition of $dF_p$ to show that $dF_p = egin{bmatrix} frac{partial f_1}{partial x_1} & frac{partial f_1}{partial x_2} & frac{partial f_1}{partial x_3} \ frac{partial f_2}{partial x_1} & frac{partial f_2}{partial x_2} & frac{partial f_2}{partial x_3} end{bmatrix}$ Hint: See the proof to Proposition 7 in the Chapter 2 Appendix

          Let $F: mathbb{R}^3 
ightarrow mathbb{R}^2$ be a differentiable function. Let $p in mathbb{R}^3$. Use the definition of $dF_p$ to show that

$dF_p = egin{bmatrix} frac{partial f_1}{partial x_1} & frac{partial f_1}{partial x_2} & frac{partial f_1}{partial x_3} \ frac{partial f_2}{partial x_1} & frac{partial f_2}{partial x_2} & frac{partial f_2}{partial x_3} end{bmatrix}$

Hint: See the proof to Proposition 7 in the Chapter 2 Appendix

$Let F: mathbbR^3 ightarrow mathbbR^2 be a differentiable function. Let p in mathbbR^3. Use the definition of dFp to show that dFp = eginbmatrix fracpartial f1partial x1 fracpartial f1partial x2 fracpartial f1partial x3 fracpartial f2partial x1 fracpartial f2partial x2 fracpartial f2partial x3 endbmatrix Hint: See the proof to Proposition 7 in the Chapter 2 Appendix$

Added by Ashley M.

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