Vector Calculus Review

12.1: Coordinate Planes and Sphere Equations

• The XY, YZ, and XZ planes divide three-dimensional space into 8 cells, commonly referred to as octants.

• The equation for the distance between two points in three-dimensional space is based on the equation for a sphere.

12.2: Vectors

• Vectors: Quantities characterized by both magnitude and direction.

• Magnitude of vectors: The length of the vector.

• Unit vectors: A vector with a magnitude of 1 in a specified direction.

12.3: Dot Product and Angle Between Vectors

• Dot product of vectors: Calculated by multiplying each component of the vectors and summing the results.

• To find the angle between two vectors u and v, use the formula:
  $\theta = ext{cos}^{-1}\left(\frac{u \cdot v}{|u||v|}\right)$

• If the dot product of two vectors equals 0, the vectors are orthogonal (perpendicular).

• Vector projection of u onto v is given by:
  $\text{proj}_{v} u = v\left(\frac{u \cdot v}{|v|^{2}}\right)$

• The scalar component of u in the direction of v is:
  $|u| \cos \Theta = \frac{u \cdot v}{|v|}$

12.4: Cross Product

• The cross product is denoted as u x v and can be calculated using the matrix method.

• In geometry, the cross product gives the area of a parallelogram formed by the two vectors. Dividing by 2 yields the area of a triangle.

• The triple scalar product (box product) is defined as $(u x v) \cdot w$

  • This gives the volume of a parallelepiped formed by vectors u, v, and w.

12.5: Parametric Equations and Planes

• If L is a line passing through a point and is parallel to vector v, the parametric equation for L is expressed as:
  $(x,y,z) = (x<em>0, y</em>0, z<em>0) + t(v</em>1, v<em>2, v</em>3)$
  where (x0, y0, z0) is the point on the line and (v1, v2, v3) are the components of vector v.

• The component equation for a plane through a point and perpendicular to vector n is:
  $Ax + By + Cz = D$
  where (A, B, C) are the components of vector n, and D is evaluated based on the coordinates of the point.

12.6: Surfaces and Corresponding Equations

• Different types of surfaces have specific equations which can be referred to in the notebook for detailed definitions.

13.1: Vector Functions

• The limit of a vector function is equal to the limit of its individual components.

• The derivative of a vector function is the derivative of each of its components. For example, the velocity is the derivative of the position vector.

13.2: Integral of Vector Functions

• The integral of a vector function is the integral of each of its components separately.

13.3: Arc Length of a Vector

• The arc length of a vector given by the vector function can be calculated as:
  $S = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2} + \left(\frac{dz}{dt}\right)^{2}} \, dt$

• The unit tangent vector for a smooth curve is given by:
  $T = \frac{v}{|v|}$
  where v is the velocity vector.

13.4: Curvature and Normal Vectors

• Curvature (κ) is defined as:
  $\kappa = \frac{1}{|v|}|\frac{dT}{dt}|$

• The normal vector N is calculated as:
  $N = \frac{\frac{dT}{dt}}{| rac{dT}{dt}|}$

13.5: Acceleration and Torsion

• The acceleration vector can be defined regarding the tangent and normal components:
  $a = a<em>{T}T + a</em>{N}N$
  where
  $a<em>{T} = \frac{d}{dt}|v|$ and $a</em>{N} = \sqrt{|a|^{2} - a_{T}^{2}}$

• Torsion is expressed as:
  $\tau = \frac{(r'(t) \times r''(t)) \cdot r'''(t)}{|r'(t) \times r''(t)|^{2}}$

13.6: Cylindrical Coordinates in Polar

• In cylindrical coordinates, translations to polar coordinates are expressed as:

• Position: $r = r<em>{u} u</em>{r} + zk$

  • Velocity:
    $v = r'u<em>{r} + r\theta'u</em>{\theta} + z'k$

  • Acceleration:
    $a = (r'' - r\theta'^{2})u<em>{r} + (r\theta'' + 2r'\theta')u</em>{\theta} + z''k$

14.1: Level Curves and Surfaces

• A level curve of a function f is the set of points in the plane at which
  $f(x, y) = c$

• The level surface of f is defined as the set of points where
  $f(x, y, z) = c$

14.2: Limits of Multivariable Functions

• For the limit of a multivariable function as it approaches a point:
  $\lim<em>{(x, y) \to (x</em>0, y_0)} f(x, y) = L$

  • The limit can often be taken by the method of substitution, similar to single-variable calculus.

• Two path test: If f(x, y) has different limits along two different paths (for example, along y = x and y = 2x), then the limit does not exist.

14.3: Partial Derivatives

• Partial derivatives allow one to differentiate with respect to one variable while treating all others as constants.

• The second-order partial derivatives maintain the same treatment, simplified as:
  $\frac{d^{2}f}{dx dy} = \frac{d}{dx}\left(\frac{d f}{dy}\right) \quad \text{and} \quad \frac{d^{2}f}{dy dx} = \frac{d}{dy}\left(\frac{d f}{dx}\right)$

• The mixed derivative theorem states that if f(x, y) and its partial derivatives are defined and continuous at a point (a, b), then:
  $f<em>{xy}(a, b) = f</em>{yx}(a, b)$

14.4: Chain Rule for Multivariable Functions

• When a function w depends on x, y, and z, each of which depends on t, the chain rule is expressed as:
  $\frac{dw}{dt} = f<em>{x}(x(t), y(t), z(t))x'(t) + f</em>{y}(x(t), y(t), z(t))y'(t) + f_{z}(x(t), y(t), z(t))z'(t)$

14.5: Gradient and Directional Derivatives

• The gradient vector of a function f(x, y) is defined as:
  $\nabla f = \frac{df}{dx} i + \frac{df}{dy} j$

• The directional derivative is found using the dot product of the gradient and the unit vector of interest, leading to:
  $D_{u}f = \nabla f \cdot u$

• It equals the magnitude of the gradient when evaluated in the direction of the gradient, marking where f increases most rapidly (decreases most rapidly if taken in the opposite direction).

• The tangent line to a level curve can be calculated as follows:
  $f<em>{x}(x</em>0, y<em>0)(x - x</em>0) + f<em>{y}(x</em>0, y<em>0)(y - y</em>0) = 0$

• The derivative along a path at point P is expressed as:
  $\nabla f \cdot (r(t) \cdot r'(t))$

14.6: Tangent Planes and Normal Lines

• The tangent plane to the function f at point P0 is given by:
  $f<em>{x}(P</em>0)(x - x<em>0) + f</em>{y}(P<em>0)(y - y</em>0) + f<em>{z}(P</em>0)(z - z_0) = 0$

• The normal line to the function f at P0 can be expressed as:
  $x = x<em>0 + f</em>{x}(P<em>0)t, \, y = y</em>0 + f<em>{y}(P</em>0)t, \, z = z<em>0 + f</em>{z}(P_0)t$

• A plane tangent to the surface z = f(x, y) at the point (x0, y0, f(x0, y0)) can be written as:
  $f<em>{x}(x</em>0, y<em>0)(x - x</em>0) + f<em>{y}(x</em>0, y<em>0)(y - y</em>0) - z - z_0 = 0$

• Linearization of function f(x, y) at P0 can be represented as:
  $L(x, y) = f(x<em>0, y</em>0) + f<em>{x}(x</em>0, y<em>0)(x - x</em>0) + f<em>{y}(x</em>0, y<em>0)(y - y</em>0)$

• The total differential of f is expressed as:
  $df = f<em>{x}(x</em>0, y<em>0)dx + f</em>{y}(x<em>0, y</em>0)dy$

Study Topics

• Torsion

• Polar Coordinates

• Review error formulas as needed.

Exam Topics

1. Binomial vector and torsion

2. Area of triangle using cross product given 3 points, and finding a unit vector perpendicular to the plane containing those points

3. Gradient vectors and directional derivatives, including identifying where the function f is increasing/decreasing most rapidly

4. Velocity and acceleration vectors expressed in terms of unit vectors: ur and uθ

5. Application of the chain rule with multivariable functions

6. Given two parametric lines (one expressed with t and the other with s), find their point of intersection and the plane defined by the lines

7. Solve initial value problems for r(t) utilizing first and second derivatives

8. Identify T (unit tangent vector), N (unit normal vector of the tangential vector's derivative w.r.t t), and curvature

9. Determine a point on a curve that lies a certain distance from a given reference point (arc length)

10. Understand the domain for multivariable functions and accurately sketch it

11. Write the equation for the plane tangent to a given surface at a specific point

12. Evaluate limits of multivariable functions

13. Find parametric equations for lines using position and velocity vectors at a given time t0

14. Given two points, compute the direction vector for the line connecting them and find their midpoint

15. Compute second-order partial derivatives

16. Derive a line parallel to a vector that intersects a specific point

17. Translate a given equation into its graphical representation

18. Translate verbal problem descriptions into mathematical equations (e.g., vector equations)