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PainterGuy
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<Moderator's note: Moved from a technical forum and thus no template.>
Hi
I was trying to understand the concept of gradient. I'm using Thomas's Calculus 12th Ed.
Please have a look here. Using the Definition 1, the answer came to be 3.54.
Then, I tried to attempt the same problem using Theorem 1 shown here. My attempt is shown below and the answer is 4.123. For some reason there are few mistakes in my code. For clarity you can refer to this image too.
I'm getting different answers using Definition 1 and Theorem 1. In my opinion, the answers should have been the same. Could you please guide me? Thank you.
##f(x,y)=z=x^{2}+(x\cdot y)##
##\frac{\partial z}{\partial x}=2x+y##
##\frac{\partial z}{\partial y}=x##
##\nabla f=\frac{\partial z}{\partial x}i+\frac{\partial z}{\partial y}j##
##\nabla f=\left( 2x+y\right) i+\left( x\right) j##
At point ##P_{0}=(1,2)##
##\nabla f=\left( 2x+y\right) i+\left( x\right) j=\left[ 2(1)+2\right] i+j=4i+j##
##\left( \frac{df}{ds}\right) _{u,P_{0}}=\left( \nabla f\right) _{P_{0}}\cdot u =\left\Vert \nabla f\right\Vert \left\Vert u\right\Vert \cos \theta ##
Unit vector in the direction ##4i+j##:
##\frac{4i}{4.123}+\frac{j}{4.123}=0.97i+0.243j##
##\left\Vert \nabla f\right\Vert \left\Vert u\right\Vert \cos \theta =0.97(4)+0.243(1)=3.88+0.243=4.123##
##\left( 4i+j\right) \left( \frac{1}{\root{2}\of{2}}i+\frac{1}{\root{2}\of{2}}j\right) =2.83+0.707=3.54##
Hi
I was trying to understand the concept of gradient. I'm using Thomas's Calculus 12th Ed.
Please have a look here. Using the Definition 1, the answer came to be 3.54.
Then, I tried to attempt the same problem using Theorem 1 shown here. My attempt is shown below and the answer is 4.123. For some reason there are few mistakes in my code. For clarity you can refer to this image too.
I'm getting different answers using Definition 1 and Theorem 1. In my opinion, the answers should have been the same. Could you please guide me? Thank you.
##f(x,y)=z=x^{2}+(x\cdot y)##
##\frac{\partial z}{\partial x}=2x+y##
##\frac{\partial z}{\partial y}=x##
##\nabla f=\frac{\partial z}{\partial x}i+\frac{\partial z}{\partial y}j##
##\nabla f=\left( 2x+y\right) i+\left( x\right) j##
At point ##P_{0}=(1,2)##
##\nabla f=\left( 2x+y\right) i+\left( x\right) j=\left[ 2(1)+2\right] i+j=4i+j##
##\left( \frac{df}{ds}\right) _{u,P_{0}}=\left( \nabla f\right) _{P_{0}}\cdot u =\left\Vert \nabla f\right\Vert \left\Vert u\right\Vert \cos \theta ##
Unit vector in the direction ##4i+j##:
##\frac{4i}{4.123}+\frac{j}{4.123}=0.97i+0.243j##
##\left\Vert \nabla f\right\Vert \left\Vert u\right\Vert \cos \theta =0.97(4)+0.243(1)=3.88+0.243=4.123##
##\left( 4i+j\right) \left( \frac{1}{\root{2}\of{2}}i+\frac{1}{\root{2}\of{2}}j\right) =2.83+0.707=3.54##
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