Question
(a) In what way are the theorems of Fubini and Clairaut similar?(b) If $ f(x, y) $ is continuous on $ [a, b] \times [c, d] $ and$$ g(x, y) = \int_a^x \int_c^y f(s, t)\ dt ds $$for $ a < x < b, c < y < d $, show that $ g_{xy} = g_{yx} = f(x, y) $.
Step 1
Fubini's theorem states that the order of integration does not affect the value of the double integral, while Clairaut's theorem states that the order of differentiation does not affect the value of the second order derivative. Both theorems require the function Show more…
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(a) In what way are the theorems of Fubini and Clairaut similar? (b) If $ f(x, y) $ is continuous on $ [a, b] \times [c, d] $ and $$ g(x, y) = \int_a^x \int_c^y f(s, t)\ dt ds $$ for $ a < x < b, c < y < d $, show that $ g_{xy} = g_{yx} = f(x, y) $.
$\begin{array}{l}{\text { (a) In what way are the theorems of Fubini and Clairaut }} \\ {\text { similar? }} \\ {\text { (b) If } f(x, y) \text { is continuous on }[a, b] \times[c, d] \text { and }}\end{array}$ $$g(x, y)=\int_{a}^{x} \int_{c}^{y} f(s, t) d t d s$$ for $a < x < b, c < y < d,$ show that $g_{x y}=g_{y x}=f(x, y).$
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(a) In what way are the theorems of Fubini and Clairaut similar? (b) If $f(x, y)$ is continuous on $[a, b] \times[c, d]$ and $$g(x, y)=\int_{a}^{x} \int_{c}^{y} f(s, t) d t d s$$ for $a< x < b, c < y < d,$ show that $g_{x y}=g_{y x}=f(x, y).$
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