More integrals

[jsbeckton]

Any hints on how to get these started? I'm completely stumped, I'm assuming that there is a proof fot the first one although I cannot find it in my book. And for the second one, there is only a passing mention of the property at the end of the chapter with no examples. Any help would be greatly appreciated.....thanks.

1) Let f and g be differentiable functions of three variables. Show that for and real numbers a and b:

\(\displaystyle {
\nabla} = gradient{\rm } \cr
\nabla (af + bg) = a\nabla f + b\nabla g \cr\)

2) Use the monotonicity property of the double integral to show:

\(\displaystyle \begin{array}{l}
\int {\int {\sin (x + y)dA{\rm } \le {\rm 1}} } \\
{\rm Where R = [0,1] x [0,1]} \\
\end{array}\)

 

[Gene]

1) Your book should say that if
y=af(x)
y'=af'(x)
and
the derivitive of f(x)+g(x)=f'(x)+g'(x)
Combining those two should be a proof.

2) I'm vague about what A and R are. Maybe someone else knows what you mean,

 

[jsbeckton]

In #2 the A stands for Area I think, there is supposed to be a subscript R on the second integral but teXaide wouldn't let me transfer that for some reason. The dA is there letting me know that I neet to decide the if the dx or dy comes first. The brackets are short-hand notation for the bounds on the integral, the first brackets contain the bounds for dx, and the second brackets contain the bounds for dy.

 

[galactus]

Any hints on how to get these started? I'm completely stumped, I'm assuming that there is a proof fot the first one although I cannot find it in my book. And for the second one, there is only a passing mention of the property at the end of the chapter with no examples. Any help would be greatly appreciated.....thanks.

1) Let f and g be differentiable functions of three variables. Show that for and real numbers a and b:

\(\displaystyle {
\forall = gradient{\rm (couldn't find right symbol)} \cr
\forall (af + bg) = a\forall f + b\forall g \cr}\)

If you're interested, that 'upside down triangle' used in gradient and partial derivative notation is called a 'nabla'. If you're using LaTex, you can type {\nabla}.

2) Use the monotonicity property of the double integral to show:

\(\displaystyle \begin{array}{l}
\int {\int {\sin (x + y)dA{\rm } \le {\rm 1}} } \\
{\rm Where R = [0,1] x [0,1]} \\
\end{array}\)

 

[jsbeckton]

thanks, do you know if the symbol is in texAide?

 

[galactus]

I don't know. I don't use it. I would venture a guess that it is though.

 

[pka]

In MathType it is on that tab marked: \(\displaystyle \partial \infty \ell\).

 

[jsbeckton]

thanks, no wonder I can't solve this, I can't even find the right symbols.

 

[jsbeckton]

Is number 2 making any lights go off for anyone? I was thinking that maybe it meant to do the double integral of the sin(x+y) between those bounds and then show that it is greater than or equal to 1, but I cannot tie in the monotinicity property:

\(\displaystyle \int {\int_R {} } f(x,y)dA \ge \int {\int_R {g(x,y)dA} }\)

Any clues?