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codyshs160
codyshs160
28.11.2019 • 
Mathematics

First, find the change by computing the line integral ∫c∇f⋅dr⃗ ∫c∇f⋅dr→, where cc is a curve connecting (0,0)(0,0) and (1,π/2)(1,π/2).
the simplest curve is the line segment joining these points. parameterize it:
with 0≤t≤10≤t≤1, r⃗ (t)=r→(t)= i⃗ +i→+ j⃗ j→
so that ∫c∇f⋅dr⃗ =∫10∫c∇f⋅dr→=∫01 dtdt
note that this isn't a very pleasant integral to evaluate by hand (though we could easily find a numerical estimate for it). it's easier to find ∫c∇f⋅dr⃗ ∫c∇f⋅dr→ as the sum ∫c1∇f⋅dr⃗ +∫c2∇f⋅dr⃗ ∫c1∇f⋅dr→+∫c2∇f⋅dr→, where c1c1 is the line segment from (0,0)(0,0) to (1,0)(1,0) and c2c2 is the line segment from (1,0)(1,0) to (1,π/2)(1,π/2). calculate these integrals to find the change in ff.
∫c1∇f⋅dr⃗ =∫c1∇f⋅dr→=
∫c2∇f⋅dr⃗ =∫c2∇f⋅dr→=
so that the change in f=∫c∇f⋅r⃗ =∫c1∇f⋅dr⃗ +∫c2∇f⋅dr⃗ =f=∫c∇f⋅r→=∫c1∇f⋅dr→+∫c2∇f⋅dr→=

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