PSU MATH 230 - Final Survival Sheet

1 | Key Concepts in Math 230: Final Survival Sheet Chapter 12: Vectors and 3D Space - The vector (called u) from point A (Ax, Ay, Az) to B (Bx, By, Bz) is 𝒖󰇍󰇍 = <𝐵𝑥−𝐴𝑥,𝐵𝑦−𝐴𝑦,𝐵𝑧−𝐴𝑧>. - ‖𝒖󰇍󰇍 ‖=√(𝐵𝑥−𝐴𝑥)2+(𝐵𝑦−𝐴𝑦)2+(𝐵𝑧−𝐴𝑧)2 - Equation of a sphere: 𝑟2=(𝑥−𝑥0)2+(𝑦−𝑦0)2+(𝑧−𝑧0)2 - Properties of Vectors o (𝒖󰇍󰇍 +𝒗󰇍󰇍 )=(𝒗󰇍󰇍 +𝒖󰇍󰇍 ) o (𝒖󰇍󰇍 +𝒗󰇍󰇍 )+𝒘󰇍󰇍󰇍 =𝒖󰇍󰇍 +(𝒗󰇍󰇍 +𝒘󰇍󰇍󰇍 ) o 0∗𝒖󰇍󰇍 =𝟎󰇍󰇍 = <0,0,0> o 𝒖󰇍󰇍 +𝟎󰇍󰇍 =𝒖󰇍󰇍 o 𝑐(𝒖󰇍󰇍 +𝒗󰇍󰇍 )=𝑐𝒖󰇍󰇍 +𝑐𝒗󰇍󰇍 o (𝑐1+𝑐2)𝒖󰇍󰇍 =𝑐1𝒖󰇍󰇍 +𝑐2𝒖󰇍󰇍 o 1∗𝒖󰇍󰇍 =𝒖󰇍󰇍 o 𝒖󰇍󰇍 +(−𝒖󰇍󰇍 )=𝟎󰇍󰇍 o 𝑐1(𝑐2𝒖󰇍󰇍 )=𝑐1𝑐2𝒖󰇍󰇍 - Unit Vectors o 𝒊 = <1,0,0> o 𝒋 = <0,1,0> o 𝒌󰇍󰇍 = <0,0,1> - Dot Products o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =𝑢1𝑣1+𝑢2𝑣2+𝑢3𝑣3 where u = <u1, u2, u3> and v = <v1, v2, v3> o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 ‖cos𝜃 o cos𝜃=(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )(‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 ‖⁄) - Properties of Dot Products o 𝒖󰇍󰇍 ⋅(𝒗󰇍󰇍 +𝒘󰇍󰇍󰇍 )=𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 +𝒖󰇍󰇍 ⋅𝒘󰇍󰇍󰇍 o 𝒖󰇍󰇍 ⋅(𝑐𝒗󰇍󰇍 )=𝑐(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 ) o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =𝒗󰇍󰇍 ⋅𝒖󰇍󰇍 o 𝟎󰇍󰇍 ⋅𝒖󰇍󰇍 =0 o 𝒖󰇍󰇍 ⋅𝒖󰇍󰇍 =‖𝒖󰇍󰇍 ‖𝟐 - 𝒖󰇍󰇍 ∙𝒗󰇍󰇍 =0 if and only if 𝒖󰇍󰇍 ⊥𝒗󰇍󰇍 - 𝑐𝑜𝑚𝑝𝒗󰇍󰇍 𝒖󰇍󰇍 =(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )‖𝒗󰇍󰇍 ‖⁄  scalar - 𝑝𝑟𝑜𝑗𝒗󰇍󰇍 𝒖󰇍󰇍 =[(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )‖𝒗󰇍󰇍 ‖][⁄𝒗󰇍󰇍 ‖𝒗󰇍󰇍 ‖]⁄  vector - Cross Products o 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =det|𝒊 𝒋 𝒌𝑢1𝑢2𝑢3𝑣1𝑣2𝑣3| o ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 ‖=‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 ‖sin𝜃 o 𝒊 ×𝒋 =𝒌󰇍󰇍 , 𝒋 ×𝒌󰇍󰇍 =𝒊 , 𝒌󰇍󰇍 ×𝒊 =𝒋 - Properties of Cross Products o 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =−(𝒗󰇍󰇍 ×𝒖󰇍󰇍 ) o 𝒖󰇍󰇍 ×(𝒗󰇍󰇍 +𝒘󰇍󰇍󰇍 )=𝒖󰇍󰇍 ×𝒗󰇍󰇍 +𝒖󰇍󰇍 ×𝒘󰇍󰇍󰇍 o 𝒖󰇍󰇍 ×(𝑐𝒗󰇍󰇍 )=𝑐(𝒖󰇍󰇍 ×𝒗󰇍󰇍 ) - 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =𝟎󰇍󰇍 if and only if 𝒖󰇍󰇍 ∥𝒗󰇍󰇍 - ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 ‖= the area of the parallelogram spanned by u and v u v A = ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 ‖2 | Key Concepts in Math 230: Final Survival Sheet - Triple Product o The triple product of vectors u, v, and w is 𝒖󰇍󰇍 ⋅(𝒗󰇍󰇍 ×𝒘󰇍󰇍󰇍 ). o 𝒖󰇍󰇍 ⋅(𝒗󰇍󰇍 ×𝒘󰇍󰇍󰇍 )= det|𝑢1𝑢2𝑢3𝑣1𝑣2𝑣3𝑤1𝑤2𝑤3| o ‖𝒖󰇍󰇍 ⋅(𝒗󰇍󰇍 ×𝒘󰇍󰇍󰇍 )‖= the volume of the parallelepiped spanned by u, v, and w. o 𝒖󰇍󰇍 ×(𝒗󰇍󰇍 ×𝒘󰇍󰇍󰇍 )=(𝒖󰇍󰇍 ⋅𝒘󰇍󰇍󰇍 )𝒗󰇍󰇍 −(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )𝒘󰇍󰇍󰇍 - Lines o 𝒓󰇍 (𝑡)=𝒓󰇍 0+𝑡𝒗󰇍󰇍  The parametric line through r0 in the direction v o 𝒓󰇍 (𝑡)=(1−𝑡)𝒓󰇍 0+𝑡𝒓󰇍 1 , 𝑡∈[0 ,1]  The line segment from r0 to r1 o 𝒓󰇍 (𝑡)=(1−𝑡)𝒓󰇍 0+𝑡𝒓󰇍 1 , 𝑡∈(−∞ ,∞)  The parametric line through r0 and r1 - Quadratic Surfaces o General form: 𝐴𝑥2+𝐵𝑦2+𝐶𝑧2+𝐸𝑥+𝐹𝑦+𝐺𝑧+𝐽=0 o Determining the shape (see chart) Chapter 13: Vector Functions - 𝒓󰇍 (𝑡)∈ℝ3, 𝑭󰇍󰇍 :ℝ→ℝ3 - The unit tangent vector of a graph is 𝑻󰇍󰇍 (𝑡)=𝒓󰇍 ′(𝑡)/‖𝒓󰇍 ′(𝑡)‖ - Differentiation rules o If 𝒓󰇍 (𝑡)=𝒖󰇍󰇍 (𝑡)⋅𝒗󰇍󰇍 (𝑡), 𝒓󰇍 ′(𝑡)=𝒖󰇍󰇍 ′(𝑡)⋅𝒗󰇍󰇍 (𝑡)+𝒖󰇍󰇍 (𝑡)⋅𝒗󰇍󰇍 ′(𝑡) o If 𝒓󰇍 (𝑡)=𝒖󰇍󰇍 (𝑡)×𝒗󰇍󰇍 (𝑡), 𝒓󰇍 ′(𝑡)=𝒖󰇍󰇍 ′(𝑡)×𝒗󰇍󰇍 (𝑡)+𝒖󰇍󰇍 (𝑡)×𝒗󰇍󰇍 ′(𝑡) o If 𝒓󰇍 (𝑡)=𝑐𝒖󰇍󰇍 (𝑡) where c is a constant, 𝒓󰇍 ′(𝑡)=𝑐𝒖󰇍󰇍 ′(𝑡) o If 𝒓󰇍 (𝑡)=𝒖󰇍󰇍 (𝑔(𝑡)), 𝒓󰇍 ′(𝑡)=𝒖󰇍󰇍 ′(𝑔(𝑡))𝑔′(𝑡) A, B, C quadratic terms Type Example All positive Ellipsoid 4𝑥2+𝑦2+𝑧2=1 All negative Ellipsoid −4𝑥2−𝑦2−𝑧2=−1 Non-zero, different signs Hyperboloid 4𝑥2+𝑦2−𝑧2=1 One is zero, the other two have same sign Paraboloid (elliptic) 𝑥2+𝑦2−𝑧=0 One is zero, the other two have different signs Paraboloid (hyperbolic) 𝑥2−𝑦2+𝑧=0 u v w3 | Key Concepts in Math 230: Final Survival Sheet - The length of the curve 𝑐={ 𝒓󰇍 (𝑡) | 𝑎≤𝑡≤𝑏 } is 𝐿=∫(𝑠𝑝𝑒𝑒𝑑)(𝑠𝑙𝑖𝑐𝑒 𝑜𝑓 𝑡𝑖𝑚𝑒)𝑓𝑖𝑛𝑖𝑠ℎ𝑠𝑡𝑎𝑟𝑡=∫‖𝒓󰇍 ′(𝑡)‖𝑏𝑎𝑑𝑡=∫√(𝑥′(𝑡))2+(𝑦′(𝑡))2𝑑𝑡𝑏𝑎 - The curvature of 𝒓󰇍 (𝑡) at t0 is 𝜅=‖𝑑𝑻󰇍󰇍 𝑑𝑠‖=‖𝑻󰇍󰇍 ′(𝑡)𝒓󰇍 ′(𝑡)‖ where T is the unit tangent vector. - Velocity =𝒗󰇍󰇍 (𝑡)=𝒓󰇍 ′(𝑡) - Acceleration =𝒂󰇍󰇍 (𝑡)=𝒓󰇍 ′′(𝑡) Chapter 14: Partial Derivatives - The partial derivative of f(x,y) is a derivative taken with one variable treated as a constant; if 𝑔(𝑥)=𝑓(𝑥,𝑐) where c is a constant, then 𝑔′(𝑥)=𝜕𝑓 𝜕𝑥⁄. o Notation: 𝜕𝑓 𝜕𝑥⁄=𝑓𝑥=𝐷𝑥 - The level curves of f(x,y) are of the form 𝐶={(𝑥,𝑦) | 𝑓(𝑥,𝑦)=𝑐} where c is a constant. - The level surfaces of f(x,y,z) are of the form 𝐶={(𝑥,𝑦,𝑧) | 𝑓(𝑥,𝑦,𝑧)=𝑐} where c is a constant. - The linear approximation of f(x,y) at (x0,y0) is 𝑓(𝑥,𝑦)≈𝑓(𝑥0,𝑦0)+𝜕𝑓𝜕𝑥(𝑥0,𝑦0)(𝑥−𝑥0)+𝜕𝑓𝜕𝑦(𝑥0,𝑦0)(𝑦−𝑦0) - The gradient of f(x,y,z) is ∇𝑓(𝑥,𝑦,𝑧)= <𝑓𝑥,𝑓𝑦,𝑓𝑧> - The tangent plane of f(x,y,z) at the point