Revision: Partial Differentiation Applied Mathematics 1 BE Civil Engineering Semester 1 (FE First Year) University of Mumbai
- F Z = Tan ( Y − a X ) + ( Y − a X ) 3 2 Then Show that ∂ 2 Z ∂ X 2 = a 2 ∂ 2 Z ∂ Y 2
- If U = E X Y Z F ( X Y Z ) Where F ( X Y Z ) is an Arbitrary Function of X Y Z . Prove That: X ∂ U ∂ X + Z ∂ U ∂ Z = Y ∂ U ∂ Y + Z ∂ U ∂ Z = 2 X Y Z . U
- Find the Maximum and Minimum Values of F ( X , Y ) = X 3 + 3 X Y 2 − 15 X 2 − 15 Y 2 + 72 X
- If Z=F(X.Y). X=R Cos θ, Y=R Sinθ. Prove that ( ∂ Z ∂ X ) 2 + ( ∂ Z ∂ Y ) 2 = ( ∂ Z ∂ R ) 2 + 1 R 2 ( ∂ Z ∂ θ ) 2
- If U = F ( Y − X X Y , Z − X X Z ) , Show that X 2 ∂ U ∂ X + Y 2 ∂ U ∂ Y + Z 2 ∂ U ∂ Z = 0 .
- If U = Sin − 1 ( X + Y √ X + √ Y ) ,Prove that X 2 U X X + 2 X Y U X Y + Y 2 U Y Y = − Sin U . Cos 2 U 4 Cos 3 U
- State and Prove Euler’S Theorem for Three Variables.
- State and Prove Euler’S Theorem for Three Variables.
- If Z = F (X, Y) Where X = Eu +E-v, Y = E-u - Ev Then Prove that ∂ Z ∂ U − ∂ Z ∂ V = X ∂ Z ∂ X − Y ∂ Z ∂ Y .
- If U = Sin − 1 [ X 1 3 + Y 1 3 X 1 2 + Y 1 2 ] Prove Hat X 2 ∂ 2 U ∂ 2 X + 2 X Y ∂ 2 U ∂ X ∂ Y ∂ 2 U ∂ 2 Y = Tan U 144 [ Tan 2 U + 13 ] .
- If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z Prove that
- If Tan(θ+Iφ)=Tanα+Isecα Prove that 1) E 2 φ = Cot ( φ 2 ) 2) 2 θ = N π + π 2 + α
- State Euler’S Theorem on Homogeneous Function of Two Variables and If U = X + Y X 2 + Y 2 Then Evaluate X ∂ U ∂ X + Y ∂ U ∂ Y
