In yoga for strength an important consideration for smooth (and relatively effortless) advancement is the idea of progressions, gradually making an exercise more difficult.
In weight lifting the equivalent would be gradually adding more and more weight.
A prerequisite for both strength and flexibility is stability.
Stability generally requires using muscles against each other with the result that the opposing muscles are strengthened. And so practicing stabilizing parts of the body is actually a strengthening exercise in and of itself.
But in general stability is the practice of making a part of the body resistant to change. Then it can be used as a foundation for another part of the body to create change.
For more on stability (and strength) you can read exercises to improve stabilty.
One way to improve strength while doing yoga poses is to practice moving in and out of yoga poses, particular those where the arms or legs bear the weight of your body.
One of the harder exercises for the arms is lifting up into wheel pose.
You can practice holding this pose to develop strength or practice lifting and lowering repeatedly.
Details on learning how to lift the head are covered in the pdf Making Wheel Pose Easier.
For strengthening the legs, one of the more difficult exercises is the one legged squat or "pistol" squat. To work up to this, first work at doing a regular squat on both legs.
Another useful preparation exercise is the seated get up knee exercise, demo'd in this video. Note the possible importance of activating the calf muscles and the back of the knee for knee stability.
From standing, see if you can lower to half way (bottom left), then stand up again. From a full deep squat (below right) you can also lift into the half squat.
In both the deep squat and the half squat you can then work at shifting weight on to one leg. Work towards this gradually. Then once you can get all of your weight onto one leg see if you can lift the other leg.
In a deep squat, if you can get all of your weight onto one leg, then reach the other leg forwards.
You can first practice standing up from this position, then lower using both legs. Once you are comfortable standing up with a once legged squat, then work at lowering down on one leg.
A pose that can be used to work on the back of the legs and or the arms is reverse plank.
To strengthen the hamstrings try these hamstring strengthening yoga poses.
Hip strengthening exercises uses some of single leg standing exercises to strengthen the hips (and knees) and improve pelvic awareness and control (relative to the thigh).
Knee strengthening exercises aren't so much about particular poses as it is about activating the knees in a variety of poses to make the knees stronger.
The back can be divided into an upper and lower half. Generally the upper back is the back of the ribcage while the lower back is the part that connects the pelvis to the ribcage (the lumbar spine.)
Upper back exercises include exercises for becoming more aware of the thoracic spine and ribcage and also training the spinal erectors to bend the ribcage backwards.
Low Back exercises (for dealing with low back pain and helping to prevent it) include a mix of low back strengthening and hip strengthening exercises.
One of the simplest exercises for the abs is to slowly sit up and balance on the butt from a laying down position. With arms and legs reaching up, slowly reach arms and legs forwards. Then when you feel your center over your buttocks, slowly sit up.
To make it easier grab onto your thighs.
Then lay back down slowly and in control and repeat.
A progression is to roll up to a squat. (First do it fast to get into the squat, then see if you can slowly roll up into a squat.)
Then you could also try rolling up into a one legged squat as above.
The seated get up is a way of getting into the one legged squat from a seated position. Even if you aren't interested in one leg squats this video does include tips on stabilizing the knees (at about the 5 minute mark.) Usual muscle activations for knee stability might include the quads, the hamstrings or any of the glutes. This looks at another set of muscles all together. If you like the video or find it helpful, please do share it! Thanks!
Some tips for learning how to do deep squats (without weight). The first tip is on how to stay balanced while squatting.
How do you learn the body weight safely? How do you work towards this pose even if you aren't sure if you are capable of doing it.
Basic yoga poses: standing, sitting, arm balances, binds, twists, inversions, back bending, front bends.
Scapular stabilization becomes a little bit harder when working agains the weight of the body. It can be easier to learn if you gradually increase the amount of body weight the scapular stabilizer muscles are working againsts..
Some hip flexor strengthening exercises.
These yoga poses can be used as arm strengthening exercises.
Turn yoga poses into leg strengthening exercises using floor pressing actions, leverage and friction.
One way of finding and fixing hip problems is to do standing hip strengthening exercises while balancing on one leg.
Knee anatomy for yoga teachers looks at the bones and muscles that comprise the back of the knee in simple terms.
Working towards a kneeling quadriceps stretch you first need to be able to kneel. If you have difficulty kneeling, you may find it helps to activate your quadriceps.
When doing quadriceps stretching it may help to activate and then relax your quadriceps in these standing, lunging, pigeon and supine yoga pose variations.
Some exercises and yoga poses for working towards a lying quadriceps stretch one leg at a time.
What are the benefits of the Dance of Shiva? Arm strength and balance, learning to learn and improving creativity.
This yoga routine video is designed to help you strengthen your arms and legs via the use of friction and pressure. It also teaches you how to become more aware of your body.
The transverse abdominis muscle can be broken down into three parts. Transverse abdominal exercises can thus affect the SI Joint, lumbar spine and the lower portion of the ribcage.
To improve the resiliency of your knees it can help to exercise them in a variety of positions. The following yoga poses can be used as knee strengthening exercises. The trick is to activate your knees while doing them.
Learn to consciously control your quads and hip flexors with Conscious Muscle Control: Quads and Superficial Hip Flexors. This downloadable video course teaches you how to feel and activate your quadriceps (the vastus muscles) as well as the rectus femoris, tensor fascae latae and sartorius muscles.
For any calf stretch you have to bend your ankle forwards to stretch the soleus and/or gastrocnemius. How you bend the ankle forwards can make the stretch more or less effective.
Yoga for flexibility with stretches for the hips, quads, hamstrings, glutes, psoas, shoulders and arms. These yoga stretches are designed to improve flexiblity.