New version page

PSU MATH 230 - Final Survival Sheet

Upgrade to remove ads

This preview shows page 1-2 out of 7 pages.

Save
View Full Document
Premium Document
Do you want full access? Go Premium and unlock all 7 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 7 pages.
Access to all documents
Download any document
Ad free experience

Upgrade to remove ads
Unformatted text preview:

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󰇍󰇍 = <𝐡π‘₯βˆ’π΄π‘₯,π΅π‘¦βˆ’π΄π‘¦,π΅π‘§βˆ’π΄π‘§>. - ‖𝒖󰇍󰇍 β€–=√(𝐡π‘₯βˆ’π΄π‘₯)2+(π΅π‘¦βˆ’π΄π‘¦)2+(π΅π‘§βˆ’π΄π‘§)2 - Equation of a sphere: π‘Ÿ2=(π‘₯βˆ’π‘₯0)2+(π‘¦βˆ’π‘¦0)2+(π‘§βˆ’π‘§0)2 - Properties of Vectors o (𝒖󰇍󰇍 +𝒗󰇍󰇍 )=(𝒗󰇍󰇍 +𝒖󰇍󰇍 ) o (𝒖󰇍󰇍 +𝒗󰇍󰇍 )+π’˜σ°‡σ°‡σ°‡ =𝒖󰇍󰇍 +(𝒗󰇍󰇍 +π’˜σ°‡σ°‡σ°‡ ) o 0βˆ—π’–σ°‡σ°‡ =πŸŽσ°‡σ°‡ = <0,0,0> o 𝒖󰇍󰇍 +πŸŽσ°‡σ°‡ =𝒖󰇍󰇍 o 𝑐(𝒖󰇍󰇍 +𝒗󰇍󰇍 )=𝑐𝒖󰇍󰇍 +𝑐𝒗󰇍󰇍 o (𝑐1+𝑐2)𝒖󰇍󰇍 =𝑐1𝒖󰇍󰇍 +𝑐2𝒖󰇍󰇍 o 1βˆ—π’–σ°‡σ°‡ =𝒖󰇍󰇍 o 𝒖󰇍󰇍 +(βˆ’π’–σ°‡σ°‡ )=πŸŽσ°‡σ°‡ o 𝑐1(𝑐2𝒖󰇍󰇍 )=𝑐1𝑐2𝒖󰇍󰇍 - Unit Vectors o π’Š = <1,0,0> o 𝒋 = <0,1,0> o π’Œσ°‡σ°‡ = <0,0,1> - Dot Products o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =𝑒1𝑣1+𝑒2𝑣2+𝑒3𝑣3 where u = <u1, u2, u3> and v = <v1, v2, v3> o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 β€–cosπœƒ o cosπœƒ=(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )(‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 ‖⁄) - Properties of Dot Products o 𝒖󰇍󰇍 β‹…(𝒗󰇍󰇍 +π’˜σ°‡σ°‡σ°‡ )=𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 +𝒖󰇍󰇍 β‹…π’˜σ°‡σ°‡σ°‡ o 𝒖󰇍󰇍 β‹…(𝑐𝒗󰇍󰇍 )=𝑐(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 ) o 𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 =𝒗󰇍󰇍 ⋅𝒖󰇍󰇍 o πŸŽσ°‡σ°‡ ⋅𝒖󰇍󰇍 =0 o 𝒖󰇍󰇍 ⋅𝒖󰇍󰇍 =‖𝒖󰇍󰇍 β€–πŸ - 𝒖󰇍󰇍 βˆ™π’—σ°‡σ°‡ =0 if and only if 𝒖󰇍󰇍 βŠ₯𝒗󰇍󰇍 - π‘π‘œπ‘šπ‘π’—σ°‡σ°‡ 𝒖󰇍󰇍 =(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )‖𝒗󰇍󰇍 ‖⁄ οƒ  scalar - π‘π‘Ÿπ‘œπ‘—π’—σ°‡σ°‡ 𝒖󰇍󰇍 =[(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )‖𝒗󰇍󰇍 β€–][⁄𝒗󰇍󰇍 ‖𝒗󰇍󰇍 β€–]⁄ οƒ  vector - Cross Products o 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =det|π’Š 𝒋 π’Œπ‘’1𝑒2𝑒3𝑣1𝑣2𝑣3| o ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 β€–=‖𝒖󰇍󰇍 ‖‖𝒗󰇍󰇍 β€–sinπœƒ o π’Š ×𝒋 =π’Œσ°‡σ°‡ , 𝒋 Γ—π’Œσ°‡σ°‡ =π’Š , π’Œσ°‡σ°‡ Γ—π’Š =𝒋 - Properties of Cross Products o 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =βˆ’(𝒗󰇍󰇍 ×𝒖󰇍󰇍 ) o 𝒖󰇍󰇍 Γ—(𝒗󰇍󰇍 +π’˜σ°‡σ°‡σ°‡ )=𝒖󰇍󰇍 ×𝒗󰇍󰇍 +𝒖󰇍󰇍 Γ—π’˜σ°‡σ°‡σ°‡ o 𝒖󰇍󰇍 Γ—(𝑐𝒗󰇍󰇍 )=𝑐(𝒖󰇍󰇍 ×𝒗󰇍󰇍 ) - 𝒖󰇍󰇍 ×𝒗󰇍󰇍 =πŸŽσ°‡σ°‡ if and only if 𝒖󰇍󰇍 βˆ₯𝒗󰇍󰇍 - ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 β€–= the area of the parallelogram spanned by u and v u v A = ‖𝒖󰇍󰇍 ×𝒗󰇍󰇍 β€–2 | Key Concepts in Math 230: Final Survival Sheet - Triple Product o The triple product of vectors u, v, and w is 𝒖󰇍󰇍 β‹…(𝒗󰇍󰇍 Γ—π’˜σ°‡σ°‡σ°‡ ). o 𝒖󰇍󰇍 β‹…(𝒗󰇍󰇍 Γ—π’˜σ°‡σ°‡σ°‡ )= det|𝑒1𝑒2𝑒3𝑣1𝑣2𝑣3𝑀1𝑀2𝑀3| o ‖𝒖󰇍󰇍 β‹…(𝒗󰇍󰇍 Γ—π’˜σ°‡σ°‡σ°‡ )β€–= the volume of the parallelepiped spanned by u, v, and w. o 𝒖󰇍󰇍 Γ—(𝒗󰇍󰇍 Γ—π’˜σ°‡σ°‡σ°‡ )=(𝒖󰇍󰇍 β‹…π’˜σ°‡σ°‡σ°‡ )𝒗󰇍󰇍 βˆ’(𝒖󰇍󰇍 ⋅𝒗󰇍󰇍 )π’˜σ°‡σ°‡σ°‡ - Lines o 𝒓󰇍 (𝑑)=𝒓󰇍 0+𝑑𝒗󰇍󰇍  The parametric line through r0 in the direction v o 𝒓󰇍 (𝑑)=(1βˆ’π‘‘)𝒓󰇍 0+𝑑𝒓󰇍 1 , π‘‘βˆˆ[0 ,1]  The line segment from r0 to r1 o 𝒓󰇍 (𝑑)=(1βˆ’π‘‘)𝒓󰇍 0+𝑑𝒓󰇍 1 , π‘‘βˆˆ(βˆ’βˆž ,∞)  The parametric line through r0 and r1 - Quadratic Surfaces o General form: 𝐴π‘₯2+𝐡𝑦2+𝐢𝑧2+𝐸π‘₯+𝐹𝑦+𝐺𝑧+𝐽=0 o Determining the shape (see chart) Chapter 13: Vector Functions - 𝒓󰇍 (𝑑)βˆˆβ„3, 𝑭󰇍󰇍 :ℝ→ℝ3 - The unit tangent vector of a graph is 𝑻󰇍󰇍 (𝑑)=𝒓󰇍 β€²(𝑑)/‖𝒓󰇍 β€²(𝑑)β€– - Differentiation rules o If 𝒓󰇍 (𝑑)=𝒖󰇍󰇍 (𝑑)⋅𝒗󰇍󰇍 (𝑑), 𝒓󰇍 β€²(𝑑)=𝒖󰇍󰇍 β€²(𝑑)⋅𝒗󰇍󰇍 (𝑑)+𝒖󰇍󰇍 (𝑑)⋅𝒗󰇍󰇍 β€²(𝑑) o If 𝒓󰇍 (𝑑)=𝒖󰇍󰇍 (𝑑)×𝒗󰇍󰇍 (𝑑), 𝒓󰇍 β€²(𝑑)=𝒖󰇍󰇍 β€²(𝑑)×𝒗󰇍󰇍 (𝑑)+𝒖󰇍󰇍 (𝑑)×𝒗󰇍󰇍 β€²(𝑑) o If 𝒓󰇍 (𝑑)=𝑐𝒖󰇍󰇍 (𝑑) where c is a constant, 𝒓󰇍 β€²(𝑑)=𝑐𝒖󰇍󰇍 β€²(𝑑) o If 𝒓󰇍 (𝑑)=𝒖󰇍󰇍 (𝑔(𝑑)), 𝒓󰇍 β€²(𝑑)=𝒖󰇍󰇍 β€²(𝑔(𝑑))𝑔′(𝑑) A, B, C quadratic terms Type Example All positive Ellipsoid 4π‘₯2+𝑦2+𝑧2=1 All negative Ellipsoid βˆ’4π‘₯2βˆ’π‘¦2βˆ’π‘§2=βˆ’1 Non-zero, different signs Hyperboloid 4π‘₯2+𝑦2βˆ’π‘§2=1 One is zero, the other two have same sign Paraboloid (elliptic) π‘₯2+𝑦2βˆ’π‘§=0 One is zero, the other two have different signs Paraboloid (hyperbolic) π‘₯2βˆ’π‘¦2+𝑧=0 u v w3 | Key Concepts in Math 230: Final Survival Sheet - The length of the curve 𝑐={ 𝒓󰇍 (𝑑) | π‘Žβ‰€π‘‘β‰€π‘ } is 𝐿=∫(𝑠𝑝𝑒𝑒𝑑)(𝑠𝑙𝑖𝑐𝑒 π‘œπ‘“ π‘‘π‘–π‘šπ‘’)π‘“π‘–π‘›π‘–π‘ β„Žπ‘ π‘‘π‘Žπ‘Ÿπ‘‘=βˆ«β€–π’“σ°‡ β€²(𝑑)β€–π‘π‘Žπ‘‘π‘‘=∫√(π‘₯β€²(𝑑))2+(𝑦′(𝑑))2π‘‘π‘‘π‘π‘Ž - The curvature of 𝒓󰇍 (𝑑) at t0 is πœ…=‖𝑑𝑻󰇍󰇍 𝑑𝑠‖=‖𝑻󰇍󰇍 β€²(𝑑)𝒓󰇍 β€²(𝑑)β€– where T is the unit tangent vector. - Velocity =𝒗󰇍󰇍 (𝑑)=𝒓󰇍 β€²(𝑑) - Acceleration =𝒂󰇍󰇍 (𝑑)=𝒓󰇍 β€²β€²(𝑑) Chapter 14: Partial Derivatives - The partial derivative of f(x,y) is a derivative taken with one variable treated as a constant; if 𝑔(π‘₯)=𝑓(π‘₯,𝑐) where c is a constant, then 𝑔′(π‘₯)=πœ•π‘“ πœ•π‘₯⁄. o Notation: πœ•π‘“ πœ•π‘₯⁄=𝑓π‘₯=𝐷π‘₯ - The level curves of f(x,y) are of the form 𝐢={(π‘₯,𝑦) | 𝑓(π‘₯,𝑦)=𝑐} where c is a constant. - The level surfaces of f(x,y,z) are of the form 𝐢={(π‘₯,𝑦,𝑧) | 𝑓(π‘₯,𝑦,𝑧)=𝑐} where c is a constant. - The linear approximation of f(x,y) at (x0,y0) is 𝑓(π‘₯,𝑦)β‰ˆπ‘“(π‘₯0,𝑦0)+πœ•π‘“πœ•π‘₯(π‘₯0,𝑦0)(π‘₯βˆ’π‘₯0)+πœ•π‘“πœ•π‘¦(π‘₯0,𝑦0)(π‘¦βˆ’π‘¦0) - The gradient of f(x,y,z) is βˆ‡π‘“(π‘₯,𝑦,𝑧)= <𝑓π‘₯,𝑓𝑦,𝑓𝑧> - The tangent plane of f(x,y,z) at the point


View Full Document
Download Final Survival Sheet
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Final Survival Sheet and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Final Survival Sheet 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?