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# PSU MATH 230 - Final Survival Sheet

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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 πσ°σ° = <π΅π₯βπ΄π₯,π΅π¦βπ΄π¦,π΅π§βπ΄π§>. - βπσ°σ° β=β(π΅π₯βπ΄π₯)2+(π΅π¦βπ΄π¦)2+(π΅π§βπ΄π§)2 - Equation of a sphere: π2=(π₯βπ₯0)2+(π¦βπ¦0)2+(π§βπ§0)2 - Properties of Vectors o (πσ°σ° +πσ°σ° )=(πσ°σ° +πσ°σ° ) o (πσ°σ° +πσ°σ° )+πσ°σ°σ° =πσ°σ° +(πσ°σ° +πσ°σ°σ° ) o 0βπσ°σ° =πσ°σ° = <0,0,0> o πσ°σ° +πσ°σ° =πσ°σ° o π(πσ°σ° +πσ°σ° )=ππσ°σ° +ππσ°σ° o (π1+π2)πσ°σ° =π1πσ°σ° +π2πσ°σ° o 1βπσ°σ° =πσ°σ° o πσ°σ° +(βπσ°σ° )=πσ°σ° o π1(π2πσ°σ° )=π1π2πσ°σ° - Unit Vectors o π = <1,0,0> o π = <0,1,0> o πσ°σ° = <0,0,1> - Dot Products o πσ°σ° βπσ°σ° =π’1π£1+π’2π£2+π’3π£3 where u = <u1, u2, u3> and v = <v1, v2, v3> o πσ°σ° βπσ°σ° =βπσ°σ° ββπσ°σ° βcosπ o cosπ=(πσ°σ° βπσ°σ° )(βπσ°σ° ββπσ°σ° ββ) - Properties of Dot Products o πσ°σ° β(πσ°σ° +πσ°σ°σ° )=πσ°σ° βπσ°σ° +πσ°σ° βπσ°σ°σ° o πσ°σ° β(ππσ°σ° )=π(πσ°σ° βπσ°σ° ) o πσ°σ° βπσ°σ° =πσ°σ° βπσ°σ° o πσ°σ° βπσ°σ° =0 o πσ°σ° βπσ°σ° =βπσ°σ° βπ - πσ°σ° βπσ°σ° =0 if and only if πσ°σ° β₯πσ°σ° - πππππσ°σ° πσ°σ° =(πσ°σ° βπσ°σ° )βπσ°σ° ββ ο  scalar - πππππσ°σ° πσ°σ° =[(πσ°σ° βπσ°σ° )βπσ°σ° β][βπσ°σ° βπσ°σ° β]β ο  vector - Cross Products o πσ°σ° Γπσ°σ° =det|π π ππ’1π’2π’3π£1π£2π£3| o βπσ°σ° Γπσ°σ° β=βπσ°σ° ββπσ°σ° βsinπ o π Γπ =πσ°σ° , π Γπσ°σ° =π , πσ°σ° Γπ =π - Properties of Cross Products o πσ°σ° Γπσ°σ° =β(πσ°σ° Γπσ°σ° ) o πσ°σ° Γ(πσ°σ° +πσ°σ°σ° )=πσ°σ° Γπσ°σ° +πσ°σ° Γπσ°σ°σ° o πσ°σ° Γ(ππσ°σ° )=π(πσ°σ° Γπσ°σ° ) - πσ°σ° Γπσ°σ° =πσ°σ° if and only if πσ°σ° β₯πσ°σ° - βπσ°σ° Γπσ°σ° β= the area of the parallelogram spanned by u and v u v A = βπσ°σ° Γπσ°σ° β2 | Key Concepts in Math 230: Final Survival Sheet - Triple Product o The triple product of vectors u, v, and w is πσ°σ° β(πσ°σ° Γπσ°σ°σ° ). o πσ°σ° β(πσ°σ° Γπσ°σ°σ° )= det|π’1π’2π’3π£1π£2π£3π€1π€2π€3| o βπσ°σ° β(πσ°σ° Γπσ°σ°σ° )β= the volume of the parallelepiped spanned by u, v, and w. o πσ°σ° Γ(πσ°σ° Γπσ°σ°σ° )=(πσ°σ° βπσ°σ°σ° )πσ°σ° β(πσ°σ° βπσ°σ° )πσ°σ°σ° - Lines o πσ° (π‘)=πσ° 0+π‘πσ°σ° ο§ The parametric line through r0 in the direction v o πσ° (π‘)=(1βπ‘)πσ° 0+π‘πσ° 1 , π‘β[0 ,1] ο§ The line segment from r0 to r1 o πσ° (π‘)=(1βπ‘)πσ° 0+π‘πσ° 1 , π‘β(ββ ,β) ο§ The parametric line through r0 and r1 - Quadratic Surfaces o General form: π΄π₯2+π΅π¦2+πΆπ§2+πΈπ₯+πΉπ¦+πΊπ§+π½=0 o Determining the shape (see chart) Chapter 13: Vector Functions - πσ° (π‘)ββ3, π­σ°σ° :βββ3 - The unit tangent vector of a graph is π»σ°σ° (π‘)=πσ° β²(π‘)/βπσ° β²(π‘)β - Differentiation rules o If πσ° (π‘)=πσ°σ° (π‘)βπσ°σ° (π‘), πσ° β²(π‘)=πσ°σ° β²(π‘)βπσ°σ° (π‘)+πσ°σ° (π‘)βπσ°σ° β²(π‘) o If πσ° (π‘)=πσ°σ° (π‘)Γπσ°σ° (π‘), πσ° β²(π‘)=πσ°σ° β²(π‘)Γπσ°σ° (π‘)+πσ°σ° (π‘)Γπσ°σ° β²(π‘) o If πσ° (π‘)=ππσ°σ° (π‘) where c is a constant, πσ° β²(π‘)=ππσ°σ° β²(π‘) o If πσ° (π‘)=πσ°σ° (π(π‘)), πσ° β²(π‘)=πσ°σ° β²(π(π‘))πβ²(π‘) A, B, C quadratic terms Type Example All positive Ellipsoid 4π₯2+π¦2+π§2=1 All negative Ellipsoid β4π₯2βπ¦2βπ§2=β1 Non-zero, different signs Hyperboloid 4π₯2+π¦2βπ§2=1 One is zero, the other two have same sign Paraboloid (elliptic) π₯2+π¦2βπ§=0 One is zero, the other two have different signs Paraboloid (hyperbolic) π₯2βπ¦2+π§=0 u v w3 | Key Concepts in Math 230: Final Survival Sheet - The length of the curve π={ πσ° (π‘) | πβ€π‘β€π } is πΏ=β«(π ππππ)(π ππππ ππ π‘πππ)πππππ βπ π‘πππ‘=β«βπσ° β²(π‘)βππππ‘=β«β(π₯β²(π‘))2+(π¦β²(π‘))2ππ‘ππ - The curvature of πσ° (π‘) at t0 is π=βππ»σ°σ° ππ β=βπ»σ°σ° β²(π‘)πσ° β²(π‘)β where T is the unit tangent vector. - Velocity =πσ°σ° (π‘)=πσ° β²(π‘) - Acceleration =πσ°σ° (π‘)=πσ° β²β²(π‘) Chapter 14: Partial Derivatives - The partial derivative of f(x,y) is a derivative taken with one variable treated as a constant; if π(π₯)=π(π₯,π) where c is a constant, then πβ²(π₯)=ππ ππ₯β. o Notation: ππ ππ₯β=ππ₯=π·π₯ - The level curves of f(x,y) are of the form πΆ={(π₯,π¦) | π(π₯,π¦)=π} where c is a constant. - The level surfaces of f(x,y,z) are of the form πΆ={(π₯,π¦,π§) | π(π₯,π¦,π§)=π} where c is a constant. - The linear approximation of f(x,y) at (x0,y0) is π(π₯,π¦)βπ(π₯0,π¦0)+ππππ₯(π₯0,π¦0)(π₯βπ₯0)+ππππ¦(π₯0,π¦0)(π¦βπ¦0) - The gradient of f(x,y,z) is βπ(π₯,π¦,π§)= <ππ₯,ππ¦,ππ§> - The tangent plane of f(x,y,z) at the point

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