Multivariable Calculus: Chapter 13: Topic Guide and Formulas (pgs ) * line segment notation above a variable indicates vector

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Baldwin Nash
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1 Multivariable Calculus: Chapter 13: Topic Guie an Formulas (pgs ) * line segment notation above a variable inicates vector The 3D Coorinate System: Distance Formula: (x 2 x ) ( y ) ) 2 y (z 2 z 1 use for: etermining the type of triangle to tell if lines are collinear: AB + BC = AC fining magnitue of a component form vector escribing a region in space given an equation/inequality, an vice versa linear equations: planes eg: x=10 the plane parallel to the yz plane, 10 units in front linear inequalities: half spaces eg: z y>2 half space consisting of all points above the plane going through the line z=y+2 perpenicular to the yz plane quaratic equations: represents two planes eg: z 2 =1 represents two horizontal planes parallel to the xy plane, z=1 is one unit above, z= 1 is one unit below quaratic inequalities: eg: with 2 variables cyliner; with 3 variables sphere (if asking for portion in between 2 spheres like an ege, use 2 inequalities) Equation of a Sphere: ( x h) 2 + (y k) 2 + ( z j) 2 = r 2 3D Vectors: a quantity with magnitue an irection, no position (can be move anywhere); epicte with a half arrowhea 1) Component Form: < a, b, c > with initial point at origin 2) Linear Combination of Stanar Unit Vectors: ai + bj + ck Aing Vectors: 1) graphically: a tail to hea, connect the result 2) a corresponing components Scalar Multiples constant multiplie by all components Unit Vector vector with magnitue of 1 to fin: u u Multiplying Vectors: 1) Scalar Multiplication answer is a vector 2) Dot Prouct answer is a scalar 3) Cross Prouct (use elimination metho & remember + +) answer is a vector

2 Dot Prouct Cross Prouct Parallel A = k B A X B = 0 Orthogonal A B = 0 if A = B X C A is orthogonal to both B an C Dot Prouct Application: A B = A * B cos θ Proof: Work: W = D F OR W = D * F cos θ where D=istance (in meters), F=force units: ft lbs or joules (given N & m) Projections: Scalar Projection of b onto a: comp a b = a b answer is scalar signe magnitue of the vector projection Vector Projection of b onto a: proj a b = ( a b ) a answer is vector shaow of b onto a scalar projection times unit vector of a Cross Prouct Application: A B = A * B sin θ Torque: rotational force: T = r * F sin θ θ is the angle between raius of the lever being pulle towars the force vector units: N m or ft lbs Area of a Parallelogram (½ for triangle): magnitue of cross prouct of ajacent sies Right Han Rule: put fingers line up with first vector in cross prouct an curl fingers towars secon vector your thumb is pointing towars the resultant cross prouct vector

3 Boat Problem: A boat is pulle onto shore using two ropes. If a foce of 325 N is neee, fin the magnitue of the force in each rope. Steps: 1) Reuce T1 an T2 to components eg: T1= T1 cos30 i + T1 sin30 j T2= T2 cos75 i T2 sin75 j 2) Set Sum of Horizontal Components equal to horizontal vector (keep in min sign of this vector) 3) Set Sum of Vertical Components equal to vertical vector which is 0 4) Use Substitution for T1 plug back into equation isolate T2 (using trig) 5) Solve Triple Scalar Prouct Volume of a Parallelopipe V = A ( B C ) if V=0 the point giving you those vectors are on the same plane Equations of Lines in 3D: to efine a line you nee: position (point) + irection (vector which line is parallel to) 1) Vector Value Function: r = < x 0, y 0, z 0 > + t < x, y, z > 2) Parametric Form: x = x t + x 0 ; y = y t + y 0 ; z = z t + z 0 x x 3) Symmetric Form: t = 0 y y x = 0 z z if irection=0; write variable = position (eg: ; y=2) y = 0 z Are two lines parallel, intersecting or skew? Parallel Are irection vectors scalar multiples of each other Intersecting Using parametric form, solve for t an s, oes this give you a common pt Skew if ^ gives you inconsistent answer (no solution) Equation of Planes in 3D: to efine plane you nee: position (point) + irection (vector normal to the plane) * if a vector is normal to a plane, it is also normal to the vectors in that plane if normal vector is given by < a, b, c > an point on plane given by < x 0, y 0, z 0 > a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0 OR A x + B y + C z + D = 0 1 Angle between 2 Vectors: cos θ = A B A B *

4 ax+by+cz+ Distance between a point an a plane: = a 2 +b 2 +c 2 use equation for plane as numerator, with coorinates of point plugge for variables enominator is foun using equation for plane Distance between 2 parallel planes: istance = Quaric Surfaces: How to Graph: 1) ientify surface an orientation 2) fin traces (plug in zeroes) 3) graph traces 4) connect traces/graph full figure a +b +c ; as in part of equation for plane for surface f(x,y)= x + y Use set notation when escribing omain an range Domain: write as {(x, y ) x + y 0 } Range: write as {f(x, y) f(x, y) 0 }

5 Types of Quaric Surfaces: cyliner: any 3D figure whose equation only uses 2 of the variables oriente about the missing variable like a cross section of a line repeate over an over in irectio of missing variable

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+