Across
- 2. A cone in 3D space results in circular contours that are ______ spaced and concentric.
- 7. The dot product gives the ________ of one vector onto another.
- 8. The _____ matrix of a function is a 2x2 matrix containing the partial derivatives of the function.
- 10. In finding the solutions to systems of differential equations, it’s important to find the eigenvalues, represented by the lowercase Greek letter _______.
- 12. Stokes Theorem is the line integral of a vector field over a loop is equal to its ___ through the enclosed surface.
- 14. For functions in multiple dimensions, if the graph of f “looks like” a ____ near a point, then f is differentiable at that point.
- 16. The double integral is categorized as the ______ under the graph f(x,y).
- 18. Assume that f and g have continuous first partial derivatives and that f (x, y) has an extremum at (x0, y0) subject to the constraint g(x, y) = 0. If ∇g(x0, y0) 6 = ~0, then there exists a number λ such that ∇f (x0, y0) = λ∇g(x0, y0). What does the λ represent?
- 19. If f(x,y,z) has continuous ___ order partial derivatives, then curl(∇f)=0.
- 20. Planes in the third dimension use the equation form z = mx + ny + b where m, n, and b are _____.
- 22. Which coordinates are the simplest to write a circle in?
- 24. A linear system is ___ if its state does remains bounded with time, is controllable if the input can be designed to take the system from any initial state to any final state, and is observable if its state can be recovered from its outputs.
Down
- 1. What is this equation? ∮F•dr = ∫∫[(∂F2/∂x) - (∂F1/∂x)]dxdy
- 3. The Jacobian ______ is used when making a change of variables when evaluating a multiple integral of a function over a region within its domain.
- 4. If there is a function f such that ∇f = the vector field F, then F must be path ______.
- 5. Movement in a vector field perpendicular to the vector field results in _____ work done.
- 6. Parameterization of a curve is rewriting the singular function of two variables as two functions of ____ variable, often t.
- 9. The partial derivative of a function f of two variables at a point (x0, y0) with respect to x is the _____ of the tangent line to the graph of f (x, y0) at the point x = x0.
- 11. The ______ of a function, representing the steepness at any point, can be helpful in finding tangent planes.
- 13. If f(x,y) approaches two different values for (x,y) → (a,b) along two different paths, the function is not ________.
- 15. For z(x,y) where there exist two functions x(t) and y(t), finding dz/dt by the ___ rule is given by [(∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)]
- 17. OF CHANGE A directional derivative measures the ________ of a function f in an arbitrary direction.
- 21. We can define the triple integral as the limit of the ___ of the product of the function times the volume of the rectangular solids.
- 23. Divergence can be defined as _____ per unit volume.